Measure, Integration & Real Analysis · Problem 23

Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · hilbert-spaces-norm-attainment

Problem

(a) Give an example of a Banach space

V

and a bounded linear functional

\varphi

V

such that

|\varphi(f)| < \|\varphi\| \|f\|

for all

f \in V \setminus \{0\}

. (b) Show there does not exist an example in part (a) where

V

is a Hilbert space.