Theorems · Theorem · functional analysis

IsHilbertSum.hasSum_linearIsometryEquiv_symm

∀ {ι : Type u_1} {𝕜 : Type u_2} [inst : RCLike 𝕜] {E : Type u_3} [inst_1 : NormedAddCommGroup E]
  [inst_2 : InnerProductSpace 𝕜 E] {G : ι → Type u_4} [inst_3 : (i : ι) → NormedAddCommGroup (G i)]
  [inst_4 : (i : ι) → InnerProductSpace 𝕜 (G i)] [inst_5 : CompleteSpace E] {V : (i : ι) → G i →ₗᵢ[𝕜] E}
  (hV : IsHilbertSum 𝕜 G V) (w : ↥(lp G 2)), HasSum (fun i => (V i) (↑w i)) (hV.linearIsometryEquiv.symm w)

In the canonical isometric isomorphism between a Hilbert sum E of G and lp G 2, a vector w : lp G 2 is the image of the infinite sum of the associated elements in E, and this sum indeed converges.

Defined in

Mathlib.Analysis.InnerProductSpace.l2Space

Cited by

0 results in Mathlib

Foundations

Depth 229 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpaceCompleteSpace