mathlib documentation

analysis.locally_convex.bounded

Von Neumann Boundedness #

This file defines natural or von Neumann bounded sets and proves elementary properties.

Main declarations #

Main results #

References #

def bornology.is_vonN_bounded (𝕜 : Type u_1) {E : Type u_2} [semi_normed_ring 𝕜] [has_scalar 𝕜 E] [has_zero E] [topological_space E] (s : set E) :
Prop

A set s is von Neumann bounded if every neighborhood of 0 absorbs s.

Equations
@[simp]
theorem bornology.is_vonN_bounded_empty (𝕜 : Type u_1) (E : Type u_2) [semi_normed_ring 𝕜] [has_scalar 𝕜 E] [has_zero E] [topological_space E] :
theorem bornology.is_vonN_bounded_iff {𝕜 : Type u_1} {E : Type u_2} [semi_normed_ring 𝕜] [has_scalar 𝕜 E] [has_zero E] [topological_space E] (s : set E) :
bornology.is_vonN_bounded 𝕜 s ↔ ∀ (V : set E), V ∈ nhds 0 → absorbs 𝕜 V s
theorem filter.has_basis.is_vonN_bounded_basis_iff {𝕜 : Type u_1} {E : Type u_2} {ι : Type u_3} [semi_normed_ring 𝕜] [has_scalar 𝕜 E] [has_zero E] [topological_space E] {q : ι → Prop} {s : ι → set E} {A : set E} (h : (nhds 0).has_basis q s) :
bornology.is_vonN_bounded 𝕜 A ↔ ∀ (i : ι), q i → absorbs 𝕜 (s i) A
theorem bornology.is_vonN_bounded.subset {𝕜 : Type u_1} {E : Type u_2} [semi_normed_ring 𝕜] [has_scalar 𝕜 E] [has_zero E] [topological_space E] {s₁ s₂ : set E} (h : s₁ ⊆ s₂) (hs₂ : bornology.is_vonN_bounded 𝕜 s₂) :
bornology.is_vonN_bounded 𝕜 s₁

Subsets of bounded sets are bounded.

theorem bornology.is_vonN_bounded.union {𝕜 : Type u_1} {E : Type u_2} [semi_normed_ring 𝕜] [has_scalar 𝕜 E] [has_zero E] [topological_space E] {s₁ s₂ : set E} (hs₁ : bornology.is_vonN_bounded 𝕜 s₁) (hs₂ : bornology.is_vonN_bounded 𝕜 s₂) :
bornology.is_vonN_bounded 𝕜 (s₁ ∪ s₂)

The union of two bounded sets is bounded.

theorem bornology.is_vonN_bounded.of_topological_space_le {𝕜 : Type u_1} {E : Type u_2} [semi_normed_ring 𝕜] [add_comm_group E] [module 𝕜 E] {t t' : topological_space E} (h : t ≤ t') {s : set E} (hs : bornology.is_vonN_bounded 𝕜 s) :

If a topology t' is coarser than t, then any set s that is bounded with respect to t is bounded with respect to t'.

theorem bornology.is_vonN_bounded.image {E : Type u_2} {𝕜₁ : Type u_4} {𝕜₂ : Type u_5} {F : Type u_6} [normed_division_ring 𝕜₁] [normed_division_ring 𝕜₂] [add_comm_group E] [module 𝕜₁ E] [add_comm_group F] [module 𝕜₂ F] [topological_space E] [topological_space F] {σ : 𝕜₁ →+* 𝕜₂} [ring_hom_surjective σ] [ring_hom_isometric σ] {s : set E} (hs : bornology.is_vonN_bounded 𝕜₁ s) (f : E →SL[σ] F) :
bornology.is_vonN_bounded 𝕜₂ (⇑f '' s)

A continuous linear image of a bounded set is bounded.

theorem bornology.is_vonN_bounded_singleton {𝕜 : Type u_1} {E : Type u_2} [normed_field 𝕜] [add_comm_group E] [module 𝕜 E] [topological_space E] [has_continuous_smul 𝕜 E] (x : E) :

Singletons are bounded.

theorem bornology.is_vonN_bounded_covers {𝕜 : Type u_1} {E : Type u_2} [normed_field 𝕜] [add_comm_group E] [module 𝕜 E] [topological_space E] [has_continuous_smul 𝕜 E] :

The union of all bounded set is the whole space.

def bornology.vonN_bornology (𝕜 : Type u_1) (E : Type u_2) [normed_field 𝕜] [add_comm_group E] [module 𝕜 E] [topological_space E] [has_continuous_smul 𝕜 E] :

The von Neumann bornology defined by the von Neumann bounded sets.

Note that this is not registered as an instance, in order to avoid diamonds with the metric bornology.

Equations
@[simp]
theorem bornology.is_bounded_iff_is_vonN_bounded (𝕜 : Type u_1) {E : Type u_2} [normed_field 𝕜] [add_comm_group E] [module 𝕜 E] [topological_space E] [has_continuous_smul 𝕜 E] {s : set E} :
theorem totally_bounded.is_vonN_bounded {𝕜 : Type u_1} {E : Type u_2} [nondiscrete_normed_field 𝕜] [add_comm_group E] [module 𝕜 E] [uniform_space E] [uniform_add_group E] [has_continuous_smul 𝕜 E] [regular_space E] {s : set E} (hs : totally_bounded s) :