Theorems · Theorem · functional analysis

banach_steinhaus_iSup_nnnorm

∀ {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} [inst_6 : RingHomIsometric σ₁₂] {ι : Type u_5}
  [CompleteSpace E] {g : ι → E →SL[σ₁₂] F}, (∀ (x : E), ⨆ i, ↑‖(g i) x‖₊ < ⊤) → ⨆ i, ↑‖g i‖₊ < ⊤

This version of Banach-Steinhaus is stated in terms of suprema of ↑‖·‖₊ : ℝ≥0∞ for convenience.

Defined in

Mathlib.Analysis.Normed.Operator.BanachSteinhaus

Cited by

0 results in Mathlib

Foundations

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

Assumes

SeminormedAddCommGroupSeminormedAddCommGroupNontriviallyNormedFieldNontriviallyNormedFieldNormedSpaceNormedSpaceRingHomIsometricCompleteSpace