Measure, Integration & Real Analysis · Problem 25

Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · hilbert-spaces-dual-lp-spaces

Problem

(a) Suppose that

\mu

is a finite measure,

1 \leq p \leq 2

, and

\varphi

is a bounded linear functional on

L^p(\mu)

. Prove that there exists

h \in L^{p'}(\mu)

such that

\varphi(f) = \int f h \, d\mu

for every

f \in L^p(\mu)

. (b) Same as (a), but with the hypothesis that

\mu

is a finite measure replaced by the hypothesis that

\mu

is a measure, and assume that

1 < p \leq 2

. [See 7.25, which along with this exercise shows that we can identify the dual of

L^p(\mu)

with

L^{p'}(\mu)

for

1 < p \leq 2

. See 9.42 for an extension to all

p \in (1, \infty)