Documentation

Mathlib.Analysis.Calculus.FormalMultilinearSeries

Formal multilinear series #

In this file we define FormalMultilinearSeries š•œ E F to be a family of n-multilinear maps for all n, designed to model the sequence of derivatives of a function. In other files we use this notion to define C^n functions (called contDiff in mathlib) and analytic functions.

Notations #

We use the notation E [Ɨn]ā†’L[š•œ] F for the space of continuous multilinear maps on E^n with values in F. This is the space in which the n-th derivative of a function from E to F lives.

Tags #

multilinear, formal series

def FormalMultilinearSeries (š•œ : Type u_1) (E : Type u_2) (F : Type u_3) [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] :
Type (max (max u_3 u_2) 0)

A formal multilinear series over a field š•œ, from E to F, is given by a family of multilinear maps from E^n to F for all n.

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    instance instAddCommGroupFormalMultilinearSeries {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] :
    Equations
    instance instInhabitedFormalMultilinearSeries {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] :
    Equations
    • instInhabitedFormalMultilinearSeries = { default := 0 }
    instance instModuleFormalMultilinearSeriesToAddCommMonoidInstAddCommGroupFormalMultilinearSeries {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] (š•œ' : Type u_1) [Semiring š•œ'] [Module š•œ' F] [ContinuousConstSMul š•œ' F] [SMulCommClass š•œ š•œ' F] :
    Module š•œ' (FormalMultilinearSeries š•œ E F)
    Equations
    @[simp]
    theorem FormalMultilinearSeries.zero_apply {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] (n : ā„•) :
    0 n = 0
    @[simp]
    theorem FormalMultilinearSeries.neg_apply {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] (f : FormalMultilinearSeries š•œ E F) (n : ā„•) :
    (-f) n = -f n
    theorem FormalMultilinearSeries.ext {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] {p : FormalMultilinearSeries š•œ E F} {q : FormalMultilinearSeries š•œ E F} (h : āˆ€ (n : ā„•), p n = q n) :
    p = q
    theorem FormalMultilinearSeries.ext_iff {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] {p : FormalMultilinearSeries š•œ E F} {q : FormalMultilinearSeries š•œ E F} :
    p = q ā†” āˆ€ (n : ā„•), p n = q n
    theorem FormalMultilinearSeries.ne_iff {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] {p : FormalMultilinearSeries š•œ E F} {q : FormalMultilinearSeries š•œ E F} :
    p ā‰  q ā†” āˆƒ (n : ā„•), p n ā‰  q n
    def FormalMultilinearSeries.prod {š•œ : Type u} {E : Type v} {F : Type w} {G : Type x} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] [AddCommGroup G] [Module š•œ G] [TopologicalSpace G] [TopologicalAddGroup G] [ContinuousConstSMul š•œ G] (p : FormalMultilinearSeries š•œ E F) (q : FormalMultilinearSeries š•œ E G) :
    FormalMultilinearSeries š•œ E (F Ɨ G)

    Cartesian product of two formal multilinear series (with the same field š•œ and the same source space, but possibly different target spaces).

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      def FormalMultilinearSeries.removeZero {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] (p : FormalMultilinearSeries š•œ E F) :

      Killing the zeroth coefficient in a formal multilinear series

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        @[simp]
        theorem FormalMultilinearSeries.removeZero_coeff_succ {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] (p : FormalMultilinearSeries š•œ E F) (n : ā„•) :
        theorem FormalMultilinearSeries.removeZero_of_pos {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] (p : FormalMultilinearSeries š•œ E F) {n : ā„•} (h : 0 < n) :
        theorem FormalMultilinearSeries.congr {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] (p : FormalMultilinearSeries š•œ E F) {m : ā„•} {n : ā„•} {v : Fin m ā†’ E} {w : Fin n ā†’ E} (h1 : m = n) (h2 : āˆ€ (i : ā„•) (him : i < m) (hin : i < n), v { val := i, isLt := him } = w { val := i, isLt := hin }) :
        (p m) v = (p n) w

        Convenience congruence lemma stating in a dependent setting that, if the arguments to a formal multilinear series are equal, then the values are also equal.

        def FormalMultilinearSeries.compContinuousLinearMap {š•œ : Type u} {E : Type v} {F : Type w} {G : Type x} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] [AddCommGroup G] [Module š•œ G] [TopologicalSpace G] [TopologicalAddGroup G] [ContinuousConstSMul š•œ G] (p : FormalMultilinearSeries š•œ F G) (u : E ā†’L[š•œ] F) :

        Composing each term pā‚™ in a formal multilinear series with (u, ..., u) where u is a fixed continuous linear map, gives a new formal multilinear series p.compContinuousLinearMap u.

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          @[simp]
          theorem FormalMultilinearSeries.compContinuousLinearMap_apply {š•œ : Type u} {E : Type v} {F : Type w} {G : Type x} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] [AddCommGroup G] [Module š•œ G] [TopologicalSpace G] [TopologicalAddGroup G] [ContinuousConstSMul š•œ G] (p : FormalMultilinearSeries š•œ F G) (u : E ā†’L[š•œ] F) (n : ā„•) (v : Fin n ā†’ E) :
          def FormalMultilinearSeries.restrictScalars (š•œ : Type u) {š•œ' : Type u'} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] [Ring š•œ'] [SMul š•œ š•œ'] [Module š•œ' E] [ContinuousConstSMul š•œ' E] [IsScalarTower š•œ š•œ' E] [Module š•œ' F] [ContinuousConstSMul š•œ' F] [IsScalarTower š•œ š•œ' F] (p : FormalMultilinearSeries š•œ' E F) :

          Reinterpret a formal š•œ'-multilinear series as a formal š•œ-multilinear series.

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            def FormalMultilinearSeries.shift {š•œ : Type u} {E : Type v} {F : Type w} [NontriviallyNormedField š•œ] [NormedAddCommGroup E] [NormedSpace š•œ E] [NormedAddCommGroup F] [NormedSpace š•œ F] (p : FormalMultilinearSeries š•œ E F) :
            FormalMultilinearSeries š•œ E (E ā†’L[š•œ] F)

            Forgetting the zeroth term in a formal multilinear series, and interpreting the following terms as multilinear maps into E ā†’L[š•œ] F. If p is the Taylor series (HasFTaylorSeriesUpTo) of a function, then p.shift is the Taylor series of the derivative of the function. Note that the p.sum of a Taylor series p does not give the original function; for a formal multilinear series that sums to the derivative of p.sum, see HasFPowerSeriesOnBall.fderiv.

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              def FormalMultilinearSeries.unshift {š•œ : Type u} {E : Type v} {F : Type w} [NontriviallyNormedField š•œ] [NormedAddCommGroup E] [NormedSpace š•œ E] [NormedAddCommGroup F] [NormedSpace š•œ F] (q : FormalMultilinearSeries š•œ E (E ā†’L[š•œ] F)) (z : F) :

              Adding a zeroth term to a formal multilinear series taking values in E ā†’L[š•œ] F. This corresponds to starting from a Taylor series (HasFTaylorSeriesUpTo) for the derivative of a function, and building a Taylor series for the function itself.

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              • One or more equations did not get rendered due to their size.
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                def ContinuousLinearMap.compFormalMultilinearSeries {š•œ : Type u} {E : Type v} {F : Type w} {G : Type x} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] [AddCommGroup G] [Module š•œ G] [TopologicalSpace G] [TopologicalAddGroup G] [ContinuousConstSMul š•œ G] (f : F ā†’L[š•œ] G) (p : FormalMultilinearSeries š•œ E F) :

                Composing each term pā‚™ in a formal multilinear series with a continuous linear map f on the left gives a new formal multilinear series f.compFormalMultilinearSeries p whose general term is f āˆ˜ pā‚™.

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                  theorem ContinuousLinearMap.compFormalMultilinearSeries_apply' {š•œ : Type u} {E : Type v} {F : Type w} {G : Type x} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] [AddCommGroup G] [Module š•œ G] [TopologicalSpace G] [TopologicalAddGroup G] [ContinuousConstSMul š•œ G] (f : F ā†’L[š•œ] G) (p : FormalMultilinearSeries š•œ E F) (n : ā„•) (v : Fin n ā†’ E) :
                  noncomputable def ContinuousMultilinearMap.toFormalMultilinearSeries {š•œ : Type u} {F : Type w} [Ring š•œ] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] {Ī¹ : Type u_1} {E : Ī¹ ā†’ Type u_2} [(i : Ī¹) ā†’ AddCommGroup (E i)] [(i : Ī¹) ā†’ Module š•œ (E i)] [(i : Ī¹) ā†’ TopologicalSpace (E i)] [āˆ€ (i : Ī¹), TopologicalAddGroup (E i)] [āˆ€ (i : Ī¹), ContinuousConstSMul š•œ (E i)] [Fintype Ī¹] (f : ContinuousMultilinearMap š•œ E F) :
                  FormalMultilinearSeries š•œ ((i : Ī¹) ā†’ E i) F

                  Realize a ContinuousMultilinearMap on āˆ€ i : Ī¹, E i as the evaluation of a FormalMultilinearSeries by choosing an arbitrary identification Ī¹ ā‰ƒ Fin (Fintype.card Ī¹).

                  Equations
                  • One or more equations did not get rendered due to their size.
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                    noncomputable def FormalMultilinearSeries.order {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] (p : FormalMultilinearSeries š•œ E F) :

                    The index of the first non-zero coefficient in p (or 0 if all coefficients are zero). This is the order of the isolated zero of an analytic function f at a point if p is the Taylor series of f at that point.

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                      @[simp]
                      theorem FormalMultilinearSeries.order_eq_find {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] {p : FormalMultilinearSeries š•œ E F} [DecidablePred fun (n : ā„•) => p n ā‰  0] (hp : āˆƒ (n : ā„•), p n ā‰  0) :
                      theorem FormalMultilinearSeries.order_eq_find' {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] {p : FormalMultilinearSeries š•œ E F} [DecidablePred fun (n : ā„•) => p n ā‰  0] (hp : p ā‰  0) :
                      theorem FormalMultilinearSeries.order_eq_zero_iff {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] {p : FormalMultilinearSeries š•œ E F} (hp : p ā‰  0) :
                      theorem FormalMultilinearSeries.apply_order_ne_zero {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] {p : FormalMultilinearSeries š•œ E F} (hp : p ā‰  0) :
                      theorem FormalMultilinearSeries.apply_eq_zero_of_lt_order {š•œ : Type u} {E : Type v} {F : Type w} [Ring š•œ] {n : ā„•} [AddCommGroup E] [Module š•œ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul š•œ E] [AddCommGroup F] [Module š•œ F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul š•œ F] {p : FormalMultilinearSeries š•œ E F} (hp : n < FormalMultilinearSeries.order p) :
                      p n = 0
                      def FormalMultilinearSeries.coeff {š•œ : Type u} {E : Type v} [NontriviallyNormedField š•œ] [NormedAddCommGroup E] [NormedSpace š•œ E] (p : FormalMultilinearSeries š•œ š•œ E) (n : ā„•) :
                      E

                      The nth coefficient of p when seen as a power series.

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                        @[simp]
                        theorem FormalMultilinearSeries.apply_eq_prod_smul_coeff {š•œ : Type u} {E : Type v} [NontriviallyNormedField š•œ] [NormedAddCommGroup E] [NormedSpace š•œ E] {p : FormalMultilinearSeries š•œ š•œ E} {n : ā„•} {y : Fin n ā†’ š•œ} :
                        (p n) y = (Finset.prod Finset.univ fun (i : Fin n) => y i) ā€¢ FormalMultilinearSeries.coeff p n
                        theorem FormalMultilinearSeries.coeff_eq_zero {š•œ : Type u} {E : Type v} [NontriviallyNormedField š•œ] [NormedAddCommGroup E] [NormedSpace š•œ E] {p : FormalMultilinearSeries š•œ š•œ E} {n : ā„•} :
                        theorem FormalMultilinearSeries.apply_eq_pow_smul_coeff {š•œ : Type u} {E : Type v} [NontriviallyNormedField š•œ] [NormedAddCommGroup E] [NormedSpace š•œ E] {p : FormalMultilinearSeries š•œ š•œ E} {n : ā„•} {z : š•œ} :
                        ((p n) fun (x : Fin n) => z) = z ^ n ā€¢ FormalMultilinearSeries.coeff p n
                        noncomputable def FormalMultilinearSeries.fslope {š•œ : Type u} {E : Type v} [NontriviallyNormedField š•œ] [NormedAddCommGroup E] [NormedSpace š•œ E] (p : FormalMultilinearSeries š•œ š•œ E) :
                        FormalMultilinearSeries š•œ š•œ E

                        The formal counterpart of dslope, corresponding to the expansion of (f z - f 0) / z. If f has p as a power series, then dslope f has fslope p as a power series.

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                          @[simp]
                          theorem FormalMultilinearSeries.coeff_iterate_fslope {š•œ : Type u} {E : Type v} [NontriviallyNormedField š•œ] [NormedAddCommGroup E] [NormedSpace š•œ E] {p : FormalMultilinearSeries š•œ š•œ E} (k : ā„•) (n : ā„•) :
                          FormalMultilinearSeries.coeff (FormalMultilinearSeries.fslope^[k] p) n = FormalMultilinearSeries.coeff p (n + k)
                          def constFormalMultilinearSeries (š•œ : Type u_1) [NontriviallyNormedField š•œ] (E : Type u_2) [NormedAddCommGroup E] [NormedSpace š•œ E] [ContinuousConstSMul š•œ E] [TopologicalAddGroup E] {F : Type u_3} [NormedAddCommGroup F] [TopologicalAddGroup F] [NormedSpace š•œ F] [ContinuousConstSMul š•œ F] (c : F) :

                          The formal multilinear series where all terms of positive degree are equal to zero, and the term of degree zero is c. It is the power series expansion of the constant function equal to c everywhere.

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                            @[simp]
                            theorem constFormalMultilinearSeries_apply {š•œ : Type u} {E : Type v} {F : Type w} [NontriviallyNormedField š•œ] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace š•œ E] [NormedSpace š•œ F] {c : F} {n : ā„•} (hn : n ā‰  0) :
                            def ContinuousLinearMap.fpowerSeries {š•œ : Type u} {E : Type v} {F : Type w} [NontriviallyNormedField š•œ] [NormedAddCommGroup E] [NormedSpace š•œ E] [NormedAddCommGroup F] [NormedSpace š•œ F] (f : E ā†’L[š•œ] F) (x : E) :

                            Formal power series of a continuous linear map f : E ā†’L[š•œ] F at x : E: f y = f x + f (y - x).

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                            • One or more equations did not get rendered due to their size.
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                              @[simp]
                              theorem ContinuousLinearMap.fpowerSeries_apply_zero {š•œ : Type u} {E : Type v} {F : Type w} [NontriviallyNormedField š•œ] [NormedAddCommGroup E] [NormedSpace š•œ E] [NormedAddCommGroup F] [NormedSpace š•œ F] (f : E ā†’L[š•œ] F) (x : E) :
                              @[simp]
                              theorem ContinuousLinearMap.fpowerSeries_apply_add_two {š•œ : Type u} {E : Type v} {F : Type w} [NontriviallyNormedField š•œ] [NormedAddCommGroup E] [NormedSpace š•œ E] [NormedAddCommGroup F] [NormedSpace š•œ F] (f : E ā†’L[š•œ] F) (x : E) (n : ā„•) :