Theorems · Theorem · functional analysis

banach_steinhaus

• 1000+ list: Banach–Steinhaus theorem

∀ {E : Type u_1} {F : Type u_2} {𝕜 : Type u_3} {𝕜₂ : Type u_4} [inst : SeminormedAddCommGroup E]
  [inst_1 : SeminormedAddCommGroup F] [inst_2 : NontriviallyNormedField 𝕜] [inst_3 : NontriviallyNormedField 𝕜₂]
  [inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {ι : Type u_5}
  [CompleteSpace E] {g : ι → E →SL[σ₁₂] F}, (∀ (x : E), ∃ C, ∀ (i : ι), ‖(g i) x‖ ≤ C) → ∃ C', ∀ (i : ι), ‖g i‖ ≤ C'

This is the standard Banach-Steinhaus theorem, or Uniform Boundedness Principle. If a family of continuous linear maps from a Banach space into a normed space is pointwise bounded, then the norms of these linear maps are uniformly bounded. See also WithSeminorms.banach_steinhaus for the general statement in barrelled spaces.

Defined in

Mathlib.Analysis.Normed.Operator.BanachSteinhaus

Cited by

3 results in Mathlib

Foundations

Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SeminormedAddCommGroupSeminormedAddCommGroupNontriviallyNormedFieldNontriviallyNormedFieldNormedSpaceNormedSpaceRingHomIsometricCompleteSpace