Documentation

Mathlib.Topology.Algebra.Module.Basic

Theory of topological modules and continuous linear maps. #

We use the class ContinuousSMul for topological (semi) modules and topological vector spaces.

In this file we define continuous (semi-)linear maps, as semilinear maps between topological modules which are continuous. The set of continuous semilinear maps between the topological R₁-module M and R₂-module M₂ with respect to the RingHom σ is denoted by M →SL[σ] M₂. Plain linear maps are denoted by M →L[R] M₂ and star-linear maps by M →L⋆[R] M₂.

The corresponding notation for equivalences is M ≃SL[σ] M₂, M ≃L[R] M₂ and M ≃L⋆[R] M₂.

theorem ContinuousSMul.of_nhds_zero {R : Type u_1} {M : Type u_2} [Ring R] [TopologicalSpace R] [TopologicalSpace M] [AddCommGroup M] [Module R M] [TopologicalRing R] [TopologicalAddGroup M] (hmul : Filter.Tendsto (fun p => p.fst p.snd) (nhds 0 ×ˢ nhds 0) (nhds 0)) (hmulleft : ∀ (m : M), Filter.Tendsto (fun a => a m) (nhds 0) (nhds 0)) (hmulright : ∀ (a : R), Filter.Tendsto (fun m => a m) (nhds 0) (nhds 0)) :

If M is a topological module over R and 0 is a limit of invertible elements of R, then is the only submodule of M with a nonempty interior. This is the case, e.g., if R is a nontrivially normed field.

Let R be a topological ring such that zero is not an isolated point (e.g., a nontrivially normed field, see NormedField.punctured_nhds_neBot). Let M be a nontrivial module over R such that c • x = 0 implies c = 0 ∨ x = 0. Then M has no isolated points. We formulate this using NeBot (𝓝[≠] x).

This lemma is not an instance because Lean would need to find [ContinuousSMul ?m_1 M] with unknown ?m_1. We register this as an instance for R = ℝ in Real.punctured_nhds_module_neBot. One can also use haveI := Module.punctured_nhds_neBot R M in a proof.

theorem continuousSMul_induced {R : Type u_2} {M₁ : Type u_3} {M₂ : Type u_4} [Semiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module R M₂] [u : TopologicalSpace R] {t : TopologicalSpace M₂} [ContinuousSMul R M₂] (f : M₁ →ₗ[R] M₂) :

The span of a separable subset with respect to a separable scalar ring is again separable.

instance Submodule.continuousSMul {α : Type u_1} {β : Type u_2} [TopologicalSpace β] [TopologicalSpace α] [Semiring α] [AddCommMonoid β] [Module α β] [ContinuousSMul α β] (S : Submodule α β) :
ContinuousSMul α { x // x S }
instance Submodule.topologicalAddGroup {α : Type u_1} {β : Type u_2} [TopologicalSpace β] [Ring α] [AddCommGroup β] [Module α β] [TopologicalAddGroup β] (S : Submodule α β) :
theorem Submodule.mapsTo_smul_closure {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] (s : Submodule R M) (c : R) :
Set.MapsTo (fun x => c x) (closure s) (closure s)
theorem Submodule.smul_closure_subset {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] (s : Submodule R M) (c : R) :
c closure s closure s

The (topological-space) closure of a submodule of a topological R-module M is itself a submodule.

Instances For

    The topological closure of a closed submodule s is equal to s.

    A subspace is dense iff its topological closure is the entire space.

    A maximal proper subspace of a topological module (i.e a Submodule satisfying IsCoatom) is either closed or dense.

    theorem LinearMap.continuous_on_pi {ι : Type u_1} {R : Type u_2} {M : Type u_3} [Finite ι] [Semiring R] [TopologicalSpace R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [ContinuousAdd M] [ContinuousSMul R M] (f : (ιR) →ₗ[R] M) :
    structure ContinuousLinearMap {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (σ : R →+* S) (M : Type u_3) [TopologicalSpace M] [AddCommMonoid M] (M₂ : Type u_4) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] extends LinearMap :
    Type (max u_3 u_4)
    • toFun : MM₂
    • map_add' : ∀ (x y : M), AddHom.toFun s.toAddHom (x + y) = AddHom.toFun s.toAddHom x + AddHom.toFun s.toAddHom y
    • map_smul' : ∀ (r : R) (x : M), AddHom.toFun s.toAddHom (r x) = σ r AddHom.toFun s.toAddHom x
    • cont : Continuous s.toFun

      Continuous linear maps between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological ring R.

    Continuous linear maps between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological ring R.

    Instances For

      Continuous linear maps between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological ring R.

      Instances For

        Continuous linear maps between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological ring R.

        Instances For

          Continuous linear maps between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological ring R.

          Instances For
            class ContinuousSemilinearMapClass (F : Type u_1) {R : outParam (Type u_2)} {S : outParam (Type u_3)} [Semiring R] [Semiring S] (σ : outParam (R →+* S)) (M : outParam (Type u_4)) [TopologicalSpace M] [AddCommMonoid M] (M₂ : outParam (Type u_5)) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] extends SemilinearMapClass :
            Type (max (max u_1 u_4) u_5)
            • coe : FMM₂
            • coe_injective' : Function.Injective FunLike.coe
            • map_add : ∀ (f : F) (x y : M), f (x + y) = f x + f y
            • map_smulₛₗ : ∀ (f : F) (r : R) (x : M), f (r x) = σ r f x
            • map_continuous : ∀ (f : F), Continuous f

              Continuity

            ContinuousSemilinearMapClass F σ M M₂ asserts F is a type of bundled continuous σ-semilinear maps M → M₂. See also ContinuousLinearMapClass F R M M₂ for the case where σ is the identity map on R. A map f between an R-module and an S-module over a ring homomorphism σ : R →+* S is semilinear if it satisfies the two properties f (x + y) = f x + f y and f (c • x) = (σ c) • f x.

            Instances
              @[inline, reducible]
              abbrev ContinuousLinearMapClass (F : Type u_1) (R : outParam (Type u_2)) [Semiring R] (M : outParam (Type u_3)) [TopologicalSpace M] [AddCommMonoid M] (M₂ : outParam (Type u_4)) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module R M₂] :
              Type (max (max u_1 u_3) u_4)

              ContinuousLinearMapClass F R M M₂ asserts F is a type of bundled continuous R-linear maps M → M₂. This is an abbreviation for ContinuousSemilinearMapClass F (RingHom.id R) M M₂.

              Instances For
                structure ContinuousLinearEquiv {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (σ : R →+* S) {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : Type u_3) [TopologicalSpace M] [AddCommMonoid M] (M₂ : Type u_4) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] extends LinearEquiv :
                Type (max u_3 u_4)
                • toFun : MM₂
                • map_add' : ∀ (x y : M), AddHom.toFun s.toAddHom (x + y) = AddHom.toFun s.toAddHom x + AddHom.toFun s.toAddHom y
                • map_smul' : ∀ (r : R) (x : M), AddHom.toFun s.toAddHom (r x) = σ r AddHom.toFun s.toAddHom x
                • invFun : M₂M
                • left_inv : Function.LeftInverse s.invFun s.toFun
                • right_inv : Function.RightInverse s.invFun s.toFun
                • continuous_toFun : Continuous s.toFun

                  Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological semiring R.

                • continuous_invFun : Continuous s.invFun

                  Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological semiring R.

                Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological semiring R.

                Instances For

                  Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological semiring R.

                  Instances For

                    Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological semiring R.

                    Instances For

                      Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological semiring R.

                      Instances For
                        class ContinuousSemilinearEquivClass (F : Type u_1) {R : outParam (Type u_2)} {S : outParam (Type u_3)} [Semiring R] [Semiring S] (σ : outParam (R →+* S)) {σ' : outParam (S →+* R)} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : outParam (Type u_4)) [TopologicalSpace M] [AddCommMonoid M] (M₂ : outParam (Type u_5)) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] extends SemilinearEquivClass :
                        Type (max (max u_1 u_4) u_5)

                        ContinuousSemilinearEquivClass F σ M M₂ asserts F is a type of bundled continuous σ-semilinear equivs M → M₂. See also ContinuousLinearEquivClass F R M M₂ for the case where σ is the identity map on R. A map f between an R-module and an S-module over a ring homomorphism σ : R →+* S is semilinear if it satisfies the two properties f (x + y) = f x + f y and f (c • x) = (σ c) • f x.

                        Instances
                          @[inline, reducible]
                          abbrev ContinuousLinearEquivClass (F : Type u_1) (R : outParam (Type u_2)) [Semiring R] (M : outParam (Type u_3)) [TopologicalSpace M] [AddCommMonoid M] (M₂ : outParam (Type u_4)) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module R M₂] :
                          Type (max (max u_1 u_3) u_4)

                          ContinuousLinearEquivClass F σ M M₂ asserts F is a type of bundled continuous R-linear equivs M → M₂. This is an abbreviation for ContinuousSemilinearEquivClass F (RingHom.id R) M M₂.

                          Instances For
                            instance ContinuousSemilinearEquivClass.continuousSemilinearMapClass (F : Type u_1) {R : Type u_2} {S : Type u_3} [Semiring R] [Semiring S] (σ : R →+* S) {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : Type u_4) [TopologicalSpace M] [AddCommMonoid M] (M₂ : Type u_5) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] [s : ContinuousSemilinearEquivClass F σ M M₂] :
                            @[simp]
                            theorem linearMapOfMemClosureRangeCoe_apply {M₁ : Type u_1} {M₂ : Type u_2} {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] {σ : R →+* S} (f : M₁M₂) (hf : f closure (Set.range FunLike.coe)) :
                            def linearMapOfMemClosureRangeCoe {M₁ : Type u_1} {M₂ : Type u_2} {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] {σ : R →+* S} (f : M₁M₂) (hf : f closure (Set.range FunLike.coe)) :
                            M₁ →ₛₗ[σ] M₂

                            Constructs a bundled linear map from a function and a proof that this function belongs to the closure of the set of linear maps.

                            Instances For
                              @[simp]
                              theorem linearMapOfTendsto_apply {M₁ : Type u_1} {M₂ : Type u_2} {α : Type u_3} {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] {σ : R →+* S} {l : Filter α} (f : M₁M₂) (g : αM₁ →ₛₗ[σ] M₂) [Filter.NeBot l] (h : Filter.Tendsto (fun a x => ↑(g a) x) l (nhds f)) :
                              ↑(linearMapOfTendsto f g h) = f
                              def linearMapOfTendsto {M₁ : Type u_1} {M₂ : Type u_2} {α : Type u_3} {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] {σ : R →+* S} {l : Filter α} (f : M₁M₂) (g : αM₁ →ₛₗ[σ] M₂) [Filter.NeBot l] (h : Filter.Tendsto (fun a x => ↑(g a) x) l (nhds f)) :
                              M₁ →ₛₗ[σ] M₂

                              Construct a bundled linear map from a pointwise limit of linear maps

                              Instances For
                                theorem LinearMap.isClosed_range_coe (M₁ : Type u_1) (M₂ : Type u_2) {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] (σ : R →+* S) :
                                IsClosed (Set.range FunLike.coe)

                                Properties that hold for non-necessarily commutative semirings. #

                                instance ContinuousLinearMap.LinearMap.coe {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] :
                                Coe (M₁ →SL[σ₁₂] M₂) (M₁ →ₛₗ[σ₁₂] M₂)

                                Coerce continuous linear maps to linear maps.

                                theorem ContinuousLinearMap.coe_injective {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] :
                                Function.Injective ContinuousLinearMap.toLinearMap
                                instance ContinuousLinearMap.continuousSemilinearMapClass {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] :
                                ContinuousSemilinearMapClass (M₁ →SL[σ₁₂] M₂) σ₁₂ M₁ M₂
                                theorem ContinuousLinearMap.coe_mk {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →ₛₗ[σ₁₂] M₂) (h : Continuous f.toFun) :

                                Coerce continuous linear maps to functions.

                                @[simp]
                                theorem ContinuousLinearMap.coe_mk' {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →ₛₗ[σ₁₂] M₂) (h : Continuous f.toFun) :
                                theorem ContinuousLinearMap.continuous {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) :
                                theorem ContinuousLinearMap.uniformContinuous {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {E₁ : Type u_9} {E₂ : Type u_10} [UniformSpace E₁] [UniformSpace E₂] [AddCommGroup E₁] [AddCommGroup E₂] [Module R₁ E₁] [Module R₂ E₂] [UniformAddGroup E₁] [UniformAddGroup E₂] (f : E₁ →SL[σ₁₂] E₂) :
                                @[simp]
                                theorem ContinuousLinearMap.coe_inj {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {f : M₁ →SL[σ₁₂] M₂} {g : M₁ →SL[σ₁₂] M₂} :
                                f = g f = g
                                theorem ContinuousLinearMap.coeFn_injective {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] :
                                Function.Injective FunLike.coe
                                def ContinuousLinearMap.Simps.apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (h : M₁ →SL[σ₁₂] M₂) :
                                M₁M₂

                                See Note [custom simps projection]. We need to specify this projection explicitly in this case, because it is a composition of multiple projections.

                                Instances For
                                  def ContinuousLinearMap.Simps.coe {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (h : M₁ →SL[σ₁₂] M₂) :
                                  M₁ →ₛₗ[σ₁₂] M₂

                                  See Note [custom simps projection].

                                  Instances For
                                    theorem ContinuousLinearMap.ext {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {f : M₁ →SL[σ₁₂] M₂} {g : M₁ →SL[σ₁₂] M₂} (h : ∀ (x : M₁), f x = g x) :
                                    f = g
                                    theorem ContinuousLinearMap.ext_iff {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {f : M₁ →SL[σ₁₂] M₂} {g : M₁ →SL[σ₁₂] M₂} :
                                    f = g ∀ (x : M₁), f x = g x
                                    def ContinuousLinearMap.copy {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (f' : M₁M₂) (h : f' = f) :
                                    M₁ →SL[σ₁₂] M₂

                                    Copy of a ContinuousLinearMap with a new toFun equal to the old one. Useful to fix definitional equalities.

                                    Instances For
                                      @[simp]
                                      theorem ContinuousLinearMap.coe_copy {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (f' : M₁M₂) (h : f' = f) :
                                      theorem ContinuousLinearMap.copy_eq {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (f' : M₁M₂) (h : f' = f) :
                                      theorem ContinuousLinearMap.map_zero {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) :
                                      f 0 = 0
                                      theorem ContinuousLinearMap.map_add {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (x : M₁) (y : M₁) :
                                      f (x + y) = f x + f y
                                      theorem ContinuousLinearMap.map_smulₛₗ {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (c : R₁) (x : M₁) :
                                      f (c x) = σ₁₂ c f x
                                      theorem ContinuousLinearMap.map_smul {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] (f : M₁ →L[R₁] M₂) (c : R₁) (x : M₁) :
                                      f (c x) = c f x
                                      @[simp]
                                      theorem ContinuousLinearMap.map_smul_of_tower {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {R : Type u_9} {S : Type u_10} [Semiring S] [SMul R M₁] [Module S M₁] [SMul R M₂] [Module S M₂] [LinearMap.CompatibleSMul M₁ M₂ R S] (f : M₁ →L[S] M₂) (c : R) (x : M₁) :
                                      f (c x) = c f x
                                      @[deprecated map_sum]
                                      theorem ContinuousLinearMap.map_sum {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {ι : Type u_9} (f : M₁ →SL[σ₁₂] M₂) (s : Finset ι) (g : ιM₁) :
                                      f (Finset.sum s fun i => g i) = Finset.sum s fun i => f (g i)
                                      @[simp]
                                      theorem ContinuousLinearMap.coe_coe {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) :
                                      f = f
                                      theorem ContinuousLinearMap.ext_ring {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [TopologicalSpace R₁] {f : R₁ →L[R₁] M₁} {g : R₁ →L[R₁] M₁} (h : f 1 = g 1) :
                                      f = g
                                      theorem ContinuousLinearMap.ext_ring_iff {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [TopologicalSpace R₁] {f : R₁ →L[R₁] M₁} {g : R₁ →L[R₁] M₁} :
                                      f = g f 1 = g 1
                                      theorem ContinuousLinearMap.eqOn_closure_span {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [T2Space M₂] {s : Set M₁} {f : M₁ →SL[σ₁₂] M₂} {g : M₁ →SL[σ₁₂] M₂} (h : Set.EqOn (f) (g) s) :
                                      Set.EqOn (f) (g) (closure ↑(Submodule.span R₁ s))

                                      If two continuous linear maps are equal on a set s, then they are equal on the closure of the Submodule.span of this set.

                                      theorem ContinuousLinearMap.ext_on {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [T2Space M₂] {s : Set M₁} (hs : Dense ↑(Submodule.span R₁ s)) {f : M₁ →SL[σ₁₂] M₂} {g : M₁ →SL[σ₁₂] M₂} (h : Set.EqOn (f) (g) s) :
                                      f = g

                                      If the submodule generated by a set s is dense in the ambient module, then two continuous linear maps equal on s are equal.

                                      theorem Submodule.topologicalClosure_map {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [RingHomSurjective σ₁₂] [TopologicalSpace R₁] [TopologicalSpace R₂] [ContinuousSMul R₁ M₁] [ContinuousAdd M₁] [ContinuousSMul R₂ M₂] [ContinuousAdd M₂] (f : M₁ →SL[σ₁₂] M₂) (s : Submodule R₁ M₁) :

                                      Under a continuous linear map, the image of the TopologicalClosure of a submodule is contained in the TopologicalClosure of its image.

                                      theorem DenseRange.topologicalClosure_map_submodule {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [RingHomSurjective σ₁₂] [TopologicalSpace R₁] [TopologicalSpace R₂] [ContinuousSMul R₁ M₁] [ContinuousAdd M₁] [ContinuousSMul R₂ M₂] [ContinuousAdd M₂] {f : M₁ →SL[σ₁₂] M₂} (hf' : DenseRange f) {s : Submodule R₁ M₁} (hs : Submodule.topologicalClosure s = ) :

                                      Under a dense continuous linear map, a submodule whose TopologicalClosure is is sent to another such submodule. That is, the image of a dense set under a map with dense range is dense.

                                      instance ContinuousLinearMap.mulAction {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {S₂ : Type u_9} [Monoid S₂] [DistribMulAction S₂ M₂] [SMulCommClass R₂ S₂ M₂] [ContinuousConstSMul S₂ M₂] :
                                      MulAction S₂ (M₁ →SL[σ₁₂] M₂)
                                      theorem ContinuousLinearMap.smul_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {S₂ : Type u_9} [Monoid S₂] [DistribMulAction S₂ M₂] [SMulCommClass R₂ S₂ M₂] [ContinuousConstSMul S₂ M₂] (c : S₂) (f : M₁ →SL[σ₁₂] M₂) (x : M₁) :
                                      ↑(c f) x = c f x
                                      @[simp]
                                      theorem ContinuousLinearMap.coe_smul {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {S₂ : Type u_9} [Monoid S₂] [DistribMulAction S₂ M₂] [SMulCommClass R₂ S₂ M₂] [ContinuousConstSMul S₂ M₂] (c : S₂) (f : M₁ →SL[σ₁₂] M₂) :
                                      ↑(c f) = c f
                                      @[simp]
                                      theorem ContinuousLinearMap.coe_smul' {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {S₂ : Type u_9} [Monoid S₂] [DistribMulAction S₂ M₂] [SMulCommClass R₂ S₂ M₂] [ContinuousConstSMul S₂ M₂] (c : S₂) (f : M₁ →SL[σ₁₂] M₂) :
                                      ↑(c f) = c f
                                      instance ContinuousLinearMap.isScalarTower {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {S₂ : Type u_9} {T₂ : Type u_10} [Monoid S₂] [Monoid T₂] [DistribMulAction S₂ M₂] [SMulCommClass R₂ S₂ M₂] [ContinuousConstSMul S₂ M₂] [DistribMulAction T₂ M₂] [SMulCommClass R₂ T₂ M₂] [ContinuousConstSMul T₂ M₂] [SMul S₂ T₂] [IsScalarTower S₂ T₂ M₂] :
                                      IsScalarTower S₂ T₂ (M₁ →SL[σ₁₂] M₂)
                                      instance ContinuousLinearMap.smulCommClass {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {S₂ : Type u_9} {T₂ : Type u_10} [Monoid S₂] [Monoid T₂] [DistribMulAction S₂ M₂] [SMulCommClass R₂ S₂ M₂] [ContinuousConstSMul S₂ M₂] [DistribMulAction T₂ M₂] [SMulCommClass R₂ T₂ M₂] [ContinuousConstSMul T₂ M₂] [SMulCommClass S₂ T₂ M₂] :
                                      SMulCommClass S₂ T₂ (M₁ →SL[σ₁₂] M₂)
                                      instance ContinuousLinearMap.zero {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] :
                                      Zero (M₁ →SL[σ₁₂] M₂)

                                      The continuous map that is constantly zero.

                                      instance ContinuousLinearMap.inhabited {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] :
                                      Inhabited (M₁ →SL[σ₁₂] M₂)
                                      @[simp]
                                      theorem ContinuousLinearMap.default_def {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] :
                                      default = 0
                                      @[simp]
                                      theorem ContinuousLinearMap.zero_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (x : M₁) :
                                      0 x = 0
                                      @[simp]
                                      theorem ContinuousLinearMap.coe_zero {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] :
                                      0 = 0
                                      theorem ContinuousLinearMap.coe_zero' {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] :
                                      0 = 0
                                      instance ContinuousLinearMap.uniqueOfLeft {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [Subsingleton M₁] :
                                      Unique (M₁ →SL[σ₁₂] M₂)
                                      instance ContinuousLinearMap.uniqueOfRight {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [Subsingleton M₂] :
                                      Unique (M₁ →SL[σ₁₂] M₂)
                                      theorem ContinuousLinearMap.exists_ne_zero {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {f : M₁ →SL[σ₁₂] M₂} (hf : f 0) :
                                      x, f x 0
                                      def ContinuousLinearMap.id (R₁ : Type u_1) [Semiring R₁] (M₁ : Type u_4) [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] :
                                      M₁ →L[R₁] M₁

                                      the identity map as a continuous linear map.

                                      Instances For
                                        instance ContinuousLinearMap.one {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] :
                                        One (M₁ →L[R₁] M₁)
                                        theorem ContinuousLinearMap.one_def {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] :
                                        theorem ContinuousLinearMap.id_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (x : M₁) :
                                        ↑(ContinuousLinearMap.id R₁ M₁) x = x
                                        @[simp]
                                        theorem ContinuousLinearMap.coe_id {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] :
                                        ↑(ContinuousLinearMap.id R₁ M₁) = LinearMap.id
                                        @[simp]
                                        theorem ContinuousLinearMap.coe_id' {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] :
                                        ↑(ContinuousLinearMap.id R₁ M₁) = id
                                        @[simp]
                                        theorem ContinuousLinearMap.coe_eq_id {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] {f : M₁ →L[R₁] M₁} :
                                        f = LinearMap.id f = ContinuousLinearMap.id R₁ M₁
                                        @[simp]
                                        theorem ContinuousLinearMap.one_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (x : M₁) :
                                        1 x = x
                                        instance ContinuousLinearMap.add {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] :
                                        Add (M₁ →SL[σ₁₂] M₂)
                                        @[simp]
                                        theorem ContinuousLinearMap.add_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] (f : M₁ →SL[σ₁₂] M₂) (g : M₁ →SL[σ₁₂] M₂) (x : M₁) :
                                        ↑(f + g) x = f x + g x
                                        @[simp]
                                        theorem ContinuousLinearMap.coe_add {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] (f : M₁ →SL[σ₁₂] M₂) (g : M₁ →SL[σ₁₂] M₂) :
                                        ↑(f + g) = f + g
                                        theorem ContinuousLinearMap.coe_add' {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] (f : M₁ →SL[σ₁₂] M₂) (g : M₁ →SL[σ₁₂] M₂) :
                                        ↑(f + g) = f + g
                                        instance ContinuousLinearMap.addCommMonoid {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] :
                                        AddCommMonoid (M₁ →SL[σ₁₂] M₂)
                                        @[simp]
                                        theorem ContinuousLinearMap.coe_sum {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] {ι : Type u_9} (t : Finset ι) (f : ιM₁ →SL[σ₁₂] M₂) :
                                        ↑(Finset.sum t fun d => f d) = Finset.sum t fun d => ↑(f d)
                                        @[simp]
                                        theorem ContinuousLinearMap.coe_sum' {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] {ι : Type u_9} (t : Finset ι) (f : ιM₁ →SL[σ₁₂] M₂) :
                                        ↑(Finset.sum t fun d => f d) = Finset.sum t fun d => ↑(f d)
                                        theorem ContinuousLinearMap.sum_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] {ι : Type u_9} (t : Finset ι) (f : ιM₁ →SL[σ₁₂] M₂) (b : M₁) :
                                        ↑(Finset.sum t fun d => f d) b = Finset.sum t fun d => ↑(f d) b
                                        def ContinuousLinearMap.comp {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) :
                                        M₁ →SL[σ₁₃] M₃

                                        Composition of bounded linear maps.

                                        Instances For

                                          Composition of bounded linear maps.

                                          Instances For
                                            @[simp]
                                            theorem ContinuousLinearMap.coe_comp {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (h : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) :
                                            @[simp]
                                            theorem ContinuousLinearMap.coe_comp' {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (h : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) :
                                            ↑(ContinuousLinearMap.comp h f) = h f
                                            theorem ContinuousLinearMap.comp_apply {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) (x : M₁) :
                                            ↑(ContinuousLinearMap.comp g f) x = g (f x)
                                            @[simp]
                                            theorem ContinuousLinearMap.comp_id {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) :
                                            @[simp]
                                            theorem ContinuousLinearMap.id_comp {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) :
                                            @[simp]
                                            theorem ContinuousLinearMap.comp_zero {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (g : M₂ →SL[σ₂₃] M₃) :
                                            @[simp]
                                            theorem ContinuousLinearMap.zero_comp {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M₁ →SL[σ₁₂] M₂) :
                                            @[simp]
                                            theorem ContinuousLinearMap.comp_add {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [ContinuousAdd M₂] [ContinuousAdd M₃] (g : M₂ →SL[σ₂₃] M₃) (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ : M₁ →SL[σ₁₂] M₂) :
                                            @[simp]
                                            theorem ContinuousLinearMap.add_comp {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [ContinuousAdd M₃] (g₁ : M₂ →SL[σ₂₃] M₃) (g₂ : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) :
                                            theorem ContinuousLinearMap.comp_assoc {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_8} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {R₄ : Type u_9} [Semiring R₄] [Module R₄ M₄] {σ₁₄ : R₁ →+* R₄} {σ₂₄ : R₂ →+* R₄} {σ₃₄ : R₃ →+* R₄} [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] [RingHomCompTriple σ₂₃ σ₃₄ σ₂₄] [RingHomCompTriple σ₁₂ σ₂₄ σ₁₄] (h : M₃ →SL[σ₃₄] M₄) (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) :
                                            instance ContinuousLinearMap.instMul {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] :
                                            Mul (M₁ →L[R₁] M₁)
                                            theorem ContinuousLinearMap.mul_def {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (f : M₁ →L[R₁] M₁) (g : M₁ →L[R₁] M₁) :
                                            @[simp]
                                            theorem ContinuousLinearMap.coe_mul {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (f : M₁ →L[R₁] M₁) (g : M₁ →L[R₁] M₁) :
                                            ↑(f * g) = f g
                                            theorem ContinuousLinearMap.mul_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (f : M₁ →L[R₁] M₁) (g : M₁ →L[R₁] M₁) (x : M₁) :
                                            ↑(f * g) x = f (g x)
                                            instance ContinuousLinearMap.monoidWithZero {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] :
                                            MonoidWithZero (M₁ →L[R₁] M₁)
                                            theorem ContinuousLinearMap.coe_pow {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (f : M₁ →L[R₁] M₁) (n : ) :
                                            ↑(f ^ n) = (f)^[n]
                                            instance ContinuousLinearMap.semiring {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] :
                                            Semiring (M₁ →L[R₁] M₁)
                                            @[simp]
                                            theorem ContinuousLinearMap.toLinearMapRingHom_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] (self : M₁ →L[R₁] M₁) :
                                            ContinuousLinearMap.toLinearMapRingHom self = self
                                            def ContinuousLinearMap.toLinearMapRingHom {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] :
                                            (M₁ →L[R₁] M₁) →+* M₁ →ₗ[R₁] M₁

                                            ContinuousLinearMap.toLinearMap as a RingHom.

                                            Instances For
                                              instance ContinuousLinearMap.applyModule {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] :
                                              Module (M₁ →L[R₁] M₁) M₁

                                              The tautological action by M₁ →L[R₁] M₁ on M.

                                              This generalizes Function.End.applyMulAction.

                                              @[simp]
                                              theorem ContinuousLinearMap.smul_def {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] (f : M₁ →L[R₁] M₁) (a : M₁) :
                                              f a = f a
                                              instance ContinuousLinearMap.applyFaithfulSMul {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] :
                                              FaithfulSMul (M₁ →L[R₁] M₁) M₁

                                              ContinuousLinearMap.applyModule is faithful.

                                              instance ContinuousLinearMap.applySMulCommClass {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] :
                                              SMulCommClass R₁ (M₁ →L[R₁] M₁) M₁
                                              instance ContinuousLinearMap.applySMulCommClass' {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] :
                                              SMulCommClass (M₁ →L[R₁] M₁) R₁ M₁
                                              instance ContinuousLinearMap.continuousConstSMul {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] :
                                              ContinuousConstSMul (M₁ →L[R₁] M₁) M₁
                                              def ContinuousLinearMap.prod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₁ →L[R₁] M₃) :
                                              M₁ →L[R₁] M₂ × M₃

                                              The cartesian product of two bounded linear maps, as a bounded linear map.

                                              Instances For
                                                @[simp]
                                                theorem ContinuousLinearMap.coe_prod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₁ →L[R₁] M₃) :
                                                ↑(ContinuousLinearMap.prod f₁ f₂) = LinearMap.prod f₁ f₂
                                                @[simp]
                                                theorem ContinuousLinearMap.prod_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₁ →L[R₁] M₃) (x : M₁) :
                                                ↑(ContinuousLinearMap.prod f₁ f₂) x = (f₁ x, f₂ x)
                                                def ContinuousLinearMap.inl (R₁ : Type u_1) [Semiring R₁] (M₁ : Type u_4) [TopologicalSpace M₁] [AddCommMonoid M₁] (M₂ : Type u_6) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] :
                                                M₁ →L[R₁] M₁ × M₂

                                                The left injection into a product is a continuous linear map.

                                                Instances For
                                                  def ContinuousLinearMap.inr (R₁ : Type u_1) [Semiring R₁] (M₁ : Type u_4) [TopologicalSpace M₁] [AddCommMonoid M₁] (M₂ : Type u_6) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] :
                                                  M₂ →L[R₁] M₁ × M₂

                                                  The right injection into a product is a continuous linear map.

                                                  Instances For
                                                    @[simp]
                                                    theorem ContinuousLinearMap.inl_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] (x : M₁) :
                                                    ↑(ContinuousLinearMap.inl R₁ M₁ M₂) x = (x, 0)
                                                    @[simp]
                                                    theorem ContinuousLinearMap.inr_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] (x : M₂) :
                                                    ↑(ContinuousLinearMap.inr R₁ M₁ M₂) x = (0, x)
                                                    @[simp]
                                                    theorem ContinuousLinearMap.coe_inl {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] :
                                                    ↑(ContinuousLinearMap.inl R₁ M₁ M₂) = LinearMap.inl R₁ M₁ M₂
                                                    @[simp]
                                                    theorem ContinuousLinearMap.coe_inr {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] :
                                                    ↑(ContinuousLinearMap.inr R₁ M₁ M₂) = LinearMap.inr R₁ M₁ M₂
                                                    theorem ContinuousLinearMap.isClosed_ker {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {F : Type u_9} [T1Space M₂] [ContinuousSemilinearMapClass F σ₁₂ M₁ M₂] (f : F) :
                                                    theorem ContinuousLinearMap.isComplete_ker {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] {F : Type u_9} {M' : Type u_10} [UniformSpace M'] [CompleteSpace M'] [AddCommMonoid M'] [Module R₁ M'] [T1Space M₂] [ContinuousSemilinearMapClass F σ₁₂ M' M₂] (f : F) :
                                                    instance ContinuousLinearMap.completeSpace_ker {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] {F : Type u_9} {M' : Type u_10} [UniformSpace M'] [CompleteSpace M'] [AddCommMonoid M'] [Module R₁ M'] [T1Space M₂] [ContinuousSemilinearMapClass F σ₁₂ M' M₂] (f : F) :
                                                    instance ContinuousLinearMap.completeSpace_eqLocus {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] {F : Type u_9} {M' : Type u_10} [UniformSpace M'] [CompleteSpace M'] [AddCommMonoid M'] [Module R₁ M'] [T2Space M₂] [ContinuousSemilinearMapClass F σ₁₂ M' M₂] (f : F) (g : F) :
                                                    @[simp]
                                                    theorem ContinuousLinearMap.ker_prod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) :
                                                    def ContinuousLinearMap.codRestrict {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ (x : M₁), f x p) :
                                                    M₁ →SL[σ₁₂] { x // x p }

                                                    Restrict codomain of a continuous linear map.

                                                    Instances For
                                                      theorem ContinuousLinearMap.coe_codRestrict {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ (x : M₁), f x p) :
                                                      @[simp]
                                                      theorem ContinuousLinearMap.coe_codRestrict_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ (x : M₁), f x p) (x : M₁) :
                                                      ↑(↑(ContinuousLinearMap.codRestrict f p h) x) = f x
                                                      @[simp]
                                                      theorem ContinuousLinearMap.ker_codRestrict {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ (x : M₁), f x p) :
                                                      def Submodule.subtypeL {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (p : Submodule R₁ M₁) :
                                                      { x // x p } →L[R₁] M₁

                                                      Submodule.subtype as a ContinuousLinearMap.

                                                      Instances For
                                                        @[simp]
                                                        theorem Submodule.coe_subtypeL {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (p : Submodule R₁ M₁) :
                                                        @[simp]
                                                        theorem Submodule.coe_subtypeL' {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (p : Submodule R₁ M₁) :
                                                        @[simp]
                                                        theorem Submodule.subtypeL_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (p : Submodule R₁ M₁) (x : { x // x p }) :
                                                        ↑(Submodule.subtypeL p) x = x
                                                        @[simp]
                                                        theorem Submodule.range_subtypeL {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (p : Submodule R₁ M₁) :
                                                        @[simp]
                                                        theorem Submodule.ker_subtypeL {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (p : Submodule R₁ M₁) :
                                                        def ContinuousLinearMap.fst (R₁ : Type u_1) [Semiring R₁] (M₁ : Type u_4) [TopologicalSpace M₁] [AddCommMonoid M₁] (M₂ : Type u_6) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] :
                                                        M₁ × M₂ →L[R₁] M₁

                                                        Prod.fst as a ContinuousLinearMap.

                                                        Instances For
                                                          def ContinuousLinearMap.snd (R₁ : Type u_1) [Semiring R₁] (M₁ : Type u_4) [TopologicalSpace M₁] [AddCommMonoid M₁] (M₂ : Type u_6) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] :
                                                          M₁ × M₂ →L[R₁] M₂

                                                          Prod.snd as a ContinuousLinearMap.

                                                          Instances For
                                                            @[simp]
                                                            theorem ContinuousLinearMap.coe_fst {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] :
                                                            ↑(ContinuousLinearMap.fst R₁ M₁ M₂) = LinearMap.fst R₁ M₁ M₂
                                                            @[simp]
                                                            theorem ContinuousLinearMap.coe_fst' {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] :
                                                            ↑(ContinuousLinearMap.fst R₁ M₁ M₂) = Prod.fst
                                                            @[simp]
                                                            theorem ContinuousLinearMap.coe_snd {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] :
                                                            ↑(ContinuousLinearMap.snd R₁ M₁ M₂) = LinearMap.snd R₁ M₁ M₂
                                                            @[simp]
                                                            theorem ContinuousLinearMap.coe_snd' {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] :
                                                            ↑(ContinuousLinearMap.snd R₁ M₁ M₂) = Prod.snd
                                                            @[simp]
                                                            theorem ContinuousLinearMap.fst_prod_snd {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] :
                                                            @[simp]
                                                            theorem ContinuousLinearMap.fst_comp_prod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) :
                                                            @[simp]
                                                            theorem ContinuousLinearMap.snd_comp_prod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) :
                                                            def ContinuousLinearMap.prodMap {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_8} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₃ →L[R₁] M₄) :
                                                            M₁ × M₃ →L[R₁] M₂ × M₄

                                                            Prod.map of two continuous linear maps.

                                                            Instances For
                                                              @[simp]
                                                              theorem ContinuousLinearMap.coe_prodMap {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_8} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₃ →L[R₁] M₄) :
                                                              ↑(ContinuousLinearMap.prodMap f₁ f₂) = LinearMap.prodMap f₁ f₂
                                                              @[simp]
                                                              theorem ContinuousLinearMap.coe_prodMap' {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_8} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₃ →L[R₁] M₄) :
                                                              ↑(ContinuousLinearMap.prodMap f₁ f₂) = Prod.map f₁ f₂
                                                              def ContinuousLinearMap.coprod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃) :
                                                              M₁ × M₂ →L[R₁] M₃

                                                              The continuous linear map given by (x, y) ↦ f₁ x + f₂ y.

                                                              Instances For
                                                                @[simp]
                                                                theorem ContinuousLinearMap.coe_coprod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃) :
                                                                ↑(ContinuousLinearMap.coprod f₁ f₂) = LinearMap.coprod f₁ f₂
                                                                @[simp]
                                                                theorem ContinuousLinearMap.coprod_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃) (x : M₁ × M₂) :
                                                                ↑(ContinuousLinearMap.coprod f₁ f₂) x = f₁ x.fst + f₂ x.snd
                                                                theorem ContinuousLinearMap.range_coprod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃) :
                                                                theorem ContinuousLinearMap.comp_fst_add_comp_snd {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f : M₁ →L[R₁] M₃) (g : M₂ →L[R₁] M₃) :
                                                                theorem ContinuousLinearMap.coprod_inl_inr {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M'₁ : Type u_5} [TopologicalSpace M'₁] [AddCommMonoid M'₁] [Module R₁ M₁] [Module R₁ M'₁] [ContinuousAdd M₁] [ContinuousAdd M'₁] :
                                                                def ContinuousLinearMap.smulRight {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {R : Type u_10} {S : Type u_11} [Semiring R] [Semiring S] [Module R M₁] [Module R M₂] [Module R S] [Module S M₂] [IsScalarTower R S M₂] [TopologicalSpace S] [ContinuousSMul S M₂] (c : M₁ →L[R] S) (f : M₂) :
                                                                M₁ →L[R] M₂

                                                                The linear map fun x => c x • f. Associates to a scalar-valued linear map and an element of M₂ the M₂-valued linear map obtained by multiplying the two (a.k.a. tensoring by M₂). See also ContinuousLinearMap.smulRightₗ and ContinuousLinearMap.smulRightL.

                                                                Instances For
                                                                  @[simp]
                                                                  theorem ContinuousLinearMap.smulRight_apply {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {R : Type u_10} {S : Type u_11} [Semiring R] [Semiring S] [Module R M₁] [Module R M₂] [Module R S] [Module S M₂] [IsScalarTower R S M₂] [TopologicalSpace S] [ContinuousSMul S M₂] {c : M₁ →L[R] S} {f : M₂} {x : M₁} :
                                                                  ↑(ContinuousLinearMap.smulRight c f) x = c x f
                                                                  @[simp]
                                                                  theorem ContinuousLinearMap.smulRight_one_one {R₁ : Type u_1} [Semiring R₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₂] [TopologicalSpace R₁] [ContinuousSMul R₁ M₂] (c : R₁ →L[R₁] M₂) :
                                                                  @[simp]
                                                                  theorem ContinuousLinearMap.smulRight_one_eq_iff {R₁ : Type u_1} [Semiring R₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₂] [TopologicalSpace R₁] [ContinuousSMul R₁ M₂] {f : M₂} {f' : M₂} :
                                                                  def ContinuousLinearMap.toSpanSingleton (R₁ : Type u_1) [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [TopologicalSpace R₁] [ContinuousSMul R₁ M₁] (x : M₁) :
                                                                  R₁ →L[R₁] M₁

                                                                  Given an element x of a topological space M over a semiring R, the natural continuous linear map from R to M by taking multiples of x.

                                                                  Instances For
                                                                    theorem ContinuousLinearMap.toSpanSingleton_apply (R₁ : Type u_1) [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [TopologicalSpace R₁] [ContinuousSMul R₁ M₁] (x : M₁) (r : R₁) :
                                                                    theorem ContinuousLinearMap.toSpanSingleton_smul' (R₁ : Type u_1) [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [TopologicalSpace R₁] [ContinuousSMul R₁ M₁] {α : Type u_10} [Monoid α] [DistribMulAction α M₁] [ContinuousConstSMul α M₁] [SMulCommClass R₁ α M₁] (c : α) (x : M₁) :

                                                                    A special case of to_span_singleton_smul' for when R is commutative.

                                                                    def ContinuousLinearMap.pi {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {ι : Type u_4} {φ : ιType u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M →L[R] φ i) :
                                                                    M →L[R] (i : ι) → φ i

                                                                    pi construction for continuous linear functions. From a family of continuous linear functions it produces a continuous linear function into a family of topological modules.

                                                                    Instances For
                                                                      @[simp]
                                                                      theorem ContinuousLinearMap.coe_pi' {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {ι : Type u_4} {φ : ιType u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M →L[R] φ i) :
                                                                      ↑(ContinuousLinearMap.pi f) = fun c i => ↑(f i) c
                                                                      @[simp]
                                                                      theorem ContinuousLinearMap.coe_pi {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {ι : Type u_4} {φ : ιType u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M →L[R] φ i) :
                                                                      ↑(ContinuousLinearMap.pi f) = LinearMap.pi fun i => ↑(f i)
                                                                      theorem ContinuousLinearMap.pi_apply {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {ι : Type u_4} {φ : ιType u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M →L[R] φ i) (c : M) (i : ι) :
                                                                      ↑(ContinuousLinearMap.pi f) c i = ↑(f i) c
                                                                      theorem ContinuousLinearMap.pi_eq_zero {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {ι : Type u_4} {φ : ιType u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M →L[R] φ i) :
                                                                      ContinuousLinearMap.pi f = 0 ∀ (i : ι), f i = 0
                                                                      theorem ContinuousLinearMap.pi_zero {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {ι : Type u_4} {φ : ιType u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] :
                                                                      (ContinuousLinearMap.pi fun x => 0) = 0
                                                                      theorem ContinuousLinearMap.pi_comp {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M₂ : Type u_3} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M₂] {ι : Type u_4} {φ : ιType u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M →L[R] φ i) (g : M₂ →L[R] M) :
                                                                      def ContinuousLinearMap.proj {R : Type u_1} [Semiring R] {ι : Type u_4} {φ : ιType u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (i : ι) :
                                                                      ((i : ι) → φ i) →L[R] φ i

                                                                      The projections from a family of topological modules are continuous linear maps.

                                                                      Instances For
                                                                        @[simp]
                                                                        theorem ContinuousLinearMap.proj_apply {R : Type u_1} [Semiring R] {ι : Type u_4} {φ : ιType u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (i : ι) (b : (i : ι) → φ i) :
                                                                        theorem ContinuousLinearMap.proj_pi {R : Type u_1} [Semiring R] {M₂ : Type u_3} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M₂] {ι : Type u_4} {φ : ιType u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M₂ →L[R] φ i) (i : ι) :
                                                                        theorem ContinuousLinearMap.iInf_ker_proj {R : Type u_1} [Semiring R] {ι : Type u_4} {φ : ιType u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] :
                                                                        def ContinuousLinearMap.iInfKerProjEquiv (R : Type u_1) [Semiring R] {ι : Type u_4} (φ : ιType u_5) [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] {I : Set ι} {J : Set ι} [DecidablePred fun i => i I] (hd : Disjoint I J) (hu : Set.univ I J) :
                                                                        { x // x ⨅ (i : ι) (_ : i J), LinearMap.ker (ContinuousLinearMap.proj i) } ≃L[R] (i : I) → φ i

                                                                        If I and J are complementary index sets, the product of the kernels of the Jth projections of φ is linearly equivalent to the product over I.

                                                                        Instances For
                                                                          theorem ContinuousLinearMap.map_neg {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →SL[σ₁₂] M₂) (x : M) :
                                                                          f (-x) = -f x
                                                                          theorem ContinuousLinearMap.map_sub {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →SL[σ₁₂] M₂) (x : M) (y : M) :
                                                                          f (x - y) = f x - f y
                                                                          @[simp]
                                                                          theorem ContinuousLinearMap.sub_apply' {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →SL[σ₁₂] M₂) (g : M →SL[σ₁₂] M₂) (x : M) :
                                                                          ↑(f - g) x = f x - g x
                                                                          theorem ContinuousLinearMap.range_prod_eq {R : Type u_1} [Ring R] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R M₂] [Module R M₃] {f : M →L[R] M₂} {g : M →L[R] M₃} (h : LinearMap.ker f LinearMap.ker g = ) :
                                                                          theorem ContinuousLinearMap.ker_prod_ker_le_ker_coprod {R : Type u_1} [Ring R] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R M₂] [Module R M₃] [ContinuousAdd M₃] (f : M →L[R] M₃) (g : M₂ →L[R] M₃) :
                                                                          instance ContinuousLinearMap.neg {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] :
                                                                          Neg (M →SL[σ₁₂] M₂)
                                                                          @[simp]
                                                                          theorem ContinuousLinearMap.neg_apply {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f : M →SL[σ₁₂] M₂) (x : M) :
                                                                          ↑(-f) x = -f x
                                                                          @[simp]
                                                                          theorem ContinuousLinearMap.coe_neg {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f : M →SL[σ₁₂] M₂) :
                                                                          ↑(-f) = -f
                                                                          theorem ContinuousLinearMap.coe_neg' {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f : M →SL[σ₁₂] M₂) :
                                                                          ↑(-f) = -f
                                                                          instance ContinuousLinearMap.sub {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] :
                                                                          Sub (M →SL[σ₁₂] M₂)
                                                                          instance ContinuousLinearMap.addCommGroup {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] :
                                                                          AddCommGroup (M →SL[σ₁₂] M₂)
                                                                          theorem ContinuousLinearMap.sub_apply {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f : M →SL[σ₁₂] M₂) (g : M →SL[σ₁₂] M₂) (x : M) :
                                                                          ↑(f - g) x = f x - g x
                                                                          @[simp]
                                                                          theorem ContinuousLinearMap.coe_sub {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f : M →SL[σ₁₂] M₂) (g : M →SL[σ₁₂] M₂) :
                                                                          ↑(f - g) = f - g
                                                                          @[simp]
                                                                          theorem ContinuousLinearMap.coe_sub' {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f : M →SL[σ₁₂] M₂) (g : M →SL[σ₁₂] M₂) :
                                                                          ↑(f - g) = f - g
                                                                          @[simp]
                                                                          theorem ContinuousLinearMap.comp_neg {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {R₃ : Type u_3} [Ring R₃] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₂] [TopologicalAddGroup M₃] (g : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) :
                                                                          @[simp]
                                                                          theorem ContinuousLinearMap.neg_comp {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {R₃ : Type u_3} [Ring R₃] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₃] (g : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) :
                                                                          @[simp]
                                                                          theorem ContinuousLinearMap.comp_sub {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {R₃ : Type u_3} [Ring R₃] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₂] [TopologicalAddGroup M₃] (g : M₂ →SL[σ₂₃] M₃) (f₁ : M →SL[σ₁₂] M₂) (f₂ : M →SL[σ₁₂] M₂) :
                                                                          @[simp]
                                                                          theorem ContinuousLinearMap.sub_comp {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {R₃ : Type u_3} [Ring R₃] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₃] (g₁ : M₂ →SL[σ₂₃] M₃) (g₂ : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) :
                                                                          def ContinuousLinearMap.projKerOfRightInverse {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse f₂ f₁) :
                                                                          M →L[R] { x // x LinearMap.ker f₁ }

                                                                          Given a right inverse f₂ : M₂ →L[R] M to f₁ : M →L[R] M₂, projKerOfRightInverse f₁ f₂ h is the projection M →L[R] LinearMap.ker f₁ along LinearMap.range f₂.

                                                                          Instances For
                                                                            @[simp]
                                                                            theorem ContinuousLinearMap.coe_projKerOfRightInverse_apply {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse f₂ f₁) (x : M) :
                                                                            ↑(↑(ContinuousLinearMap.projKerOfRightInverse f₁ f₂ h) x) = x - f₂ (f₁ x)
                                                                            @[simp]
                                                                            theorem ContinuousLinearMap.projKerOfRightInverse_apply_idem {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse f₂ f₁) (x : { x // x LinearMap.ker f₁ }) :
                                                                            @[simp]
                                                                            theorem ContinuousLinearMap.projKerOfRightInverse_comp_inv {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse f₂ f₁) (y : M₂) :
                                                                            ↑(ContinuousLinearMap.projKerOfRightInverse f₁ f₂ h) (f₂ y) = 0

                                                                            A nonzero continuous linear functional is open.

                                                                            @[simp]
                                                                            theorem ContinuousLinearMap.smul_comp {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {S₃ : Type u_5} [Semiring R] [Semiring R₂] [Semiring R₃] [Monoid S₃] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M₂ : Type u_7} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] {M₃ : Type u_8} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₃ M₃] [DistribMulAction S₃ M₃] [SMulCommClass R₃ S₃ M₃] [ContinuousConstSMul S₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (c : S₃) (h : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) :
                                                                            @[simp]
                                                                            theorem ContinuousLinearMap.comp_smul {R : Type u_1} {S : Type u_4} [Semiring R] [Monoid S] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {N₂ : Type u_9} [TopologicalSpace N₂] [AddCommMonoid N₂] [Module R N₂] {N₃ : Type u_10} [TopologicalSpace N₃] [AddCommMonoid N₃] [Module R N₃] [DistribMulAction S N₃] [SMulCommClass R S N₃] [ContinuousConstSMul S N₃] [DistribMulAction S N₂] [ContinuousConstSMul S N₂] [SMulCommClass R S N₂] [LinearMap.CompatibleSMul N₂ N₃ S R] (hₗ : N₂ →L[R] N₃) (c : S) (fₗ : M →L[R] N₂) :
                                                                            @[simp]
                                                                            theorem ContinuousLinearMap.comp_smulₛₗ {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R] [Semiring R₂] [Semiring R₃] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M₂ : Type u_7} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] {M₃ : Type u_8} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [SMulCommClass R₂ R₂ M₂] [SMulCommClass R₃ R₃ M₃] [ContinuousConstSMul R₂ M₂] [ContinuousConstSMul R₃ M₃] (h : M₂ →SL[σ₂₃] M₃) (c : R₂) (f : M →SL[σ₁₂] M₂) :
                                                                            instance ContinuousLinearMap.distribMulAction {R : Type u_1} {R₂ : Type u_2} {S₃ : Type u_5} [Semiring R] [Semiring R₂] [Monoid S₃] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M₂ : Type u_7} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [DistribMulAction S₃ M₂] [ContinuousConstSMul S₃ M₂] [SMulCommClass R₂ S₃ M₂] [ContinuousAdd M₂] :
                                                                            DistribMulAction S₃ (M →SL[σ₁₂] M₂)
                                                                            @[simp]
                                                                            theorem ContinuousLinearMap.prodEquiv_apply {R : Type u_1} [Semiring R] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {N₂ : Type u_9} [TopologicalSpace N₂] [AddCommMonoid N₂] [Module R N₂] {N₃ : Type u_10} [TopologicalSpace N₃] [AddCommMonoid N₃] [Module R N₃] (f : (M →L[R] N₂) × (M →L[R] N₃)) :
                                                                            ContinuousLinearMap.prodEquiv f = ContinuousLinearMap.prod f.fst f.snd
                                                                            def ContinuousLinearMap.prodEquiv {R : Type u_1} [Semiring R] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {N₂ : Type u_9} [TopologicalSpace N₂] [AddCommMonoid N₂] [Module R N₂] {N₃ : Type u_10} [TopologicalSpace N₃] [AddCommMonoid N₃] [Module R N₃] :
                                                                            (M →L[R] N₂) × (M →L[R] N₃) (M →L[R] N₂ × N₃)

                                                                            ContinuousLinearMap.prod as an Equiv.

                                                                            Instances For