Measure, Integration & Real Analysis · Problem 6E.18

L^p Spaces — Sheldon Axler

Mathematics — Measure Theory · weak-boundedness-implies-norm-bounded

Problem

Suppose that

V

is a normed vector space and

B

is a subset of

V

such that

\sup_{f \in B} |\varphi(f)| < \infty

for every

\varphi \in V'

. Prove that

\sup_{f \in B} \|f\| < \infty