Measure, Integration & Real Analysis · Problem 17

Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · hilbert-spaces-extension-bounded-linear-map

Problem

Suppose

U

is a subspace of a Hilbert space

V

. Suppose also that

W

is a Banach space and

S: U \to W

is a bounded linear map. Prove that there exists a bounded linear map

T: V \to W

such that

T|_U = S

and

\|T\| = \|S\|

. [If

W = \mathbf{F}

, then this result is just the Hahn--Banach Theorem (6.69) for Hilbert spaces. The result here is stronger because it allows

W

to be an arbitrary Banach space instead of requiring

W

to be

\mathbf{F}

. Also, the proof in this Hilbert space context does not require use of Zorn's Lemma or the Axiom of Choice.]