Measure, Integration & Real Analysis · Problem 6C.6

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · bounded-functions-banach-space

Problem

Suppose

X

is a nonempty set and

b(X)

is the vector space of bounded functions from

X

\mathbf{F}

. Prove that if

\|\cdot\|

is defined on

b(X)

\|f\| = \sup_X |f|

, then

b(X)

is a Banach space.