Bourbaki

Corollary

Tag 07R8, permanent. Cite it as [Bourbaki, Tag 07R8].

Topological Vector Spaces, chapter IV, § 1. Duality

— Let E˙\dot{E} be a normed space satisfying the first axiom of countability. There exists a countable subset D of E′−{0}E' - \{0\} such that we have

∥x∥=sup⁡ξ∈D∣⟨x,ξ⟩∣/∥ξ∥ \|x\| = \sup_{\xi \in D} |\langle x, \xi \rangle| / \|\xi\|

for all x∈Ex \in E.

Let B' be the unit ball of the dual E′E' of E with the weak topology σ(E′,E)\sigma(E', E) assigned to it. Then B' is a compact metrizable space (III, p. 19, cor. 2); hence there exists a countable dense subset D' in B'. Put D=D′∩(E′−{0})D = D' \cap (E' - \{0\}). Let x∈Ex \in E; the mapping x′↦⟨x,x′⟩x' \mapsto \langle x, x' \rangle from B' into K is continuous, therefore

sup⁡x′∈B′∣⟨x,x′⟩∣=sup⁡ξ∈D′∣⟨x,ξ⟩∣≤sup⁡ξ∈D∣⟨x,ξ⟩∣/∥ξ∥≤∥x∥. \sup_{x' \in B'} |\langle x, x' \rangle| = \sup_{\xi \in D'} |\langle x, \xi \rangle| \leq \sup_{\xi \in D} |\langle x, \xi \rangle| / \|\xi\| \leq \|x\|.

Formula (5) now follows from (3).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Transcribed from the printed volume. Not translated.