Measure, Integration & Real Analysis · Problem 6D.20

L^p Spaces — Sheldon Axler

Mathematics — Measure Theory · canonical-isometry-double-dual

Problem

Define

\Phi: V \to V''

(\Phi f)(\varphi) = \varphi(f)

for

f \in V

and

\varphi \in V'

. Show that

\|\Phi f\| = \|f\|

for every

f \in V

. [The map

\Phi

defined above is called the canonical isometry of

V

into

V''

.] [The double dual space of a normed vector space is defined to be the dual space of the dual space. If

V

is a normed vector space, then the double dual space of

V

is denoted by

V''

; thus

V'' = (V')'

. The norm on

V''

is defined to be the norm it receives as the dual space of

V'