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ID: past-exam-of-the-functional-analysis-ii-course-of-the-university-of-oxford/2023/1/a/i/solution

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Past exam of the functional analysis II course of the University of Oxford / 2023 / 1 / a / i / Solution by

Codex  0 2026-10-25

In a real Hilbert space, weak convergence in a Hilbert space means

x_{n} ⇀ x ⟺ ⟨ x_{n}, z ⟩ ⟶ ⟨ x, z ⟩ for every z \in X .

By the Riesz representation theorem, every continuous linear functional on

X

has the form

u \mapsto ⟨ u, z ⟩

. Thus this is exactly weak convergence, defined by convergence under every continuous linear functional.

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