Measure, Integration & Real Analysis · Problem 14

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · bounded-linear-maps-extension

Problem

Suppose

U

is a subspace of a normed vector space

V

. Suppose also that

W

is a Banach space and

S: U \to W

is a bounded linear map. (a) Prove that there exists a unique continuous function

T: \overline{U} \to W

such that

T|_U = S

. (b) Prove that the function

T

in (a) is a bounded linear map from

\overline{U}

W

and

\|T\| = \|S\|

. (c) Give an example to show that (a) can fail if the assumption that

W

is a Banach space is replaced by the assumption that

W

is a normed vector space.