Measure, Integration & Real Analysis · Problem 6C.13

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · normed-space-completion

Problem

Suppose

U

is a normed vector space. Let

d

be the metric on

U

defined by

d(f, g) = \|f - g\|

for

f, g \in U

. Let

V

be the complete metric space constructed in Exercise 16 in Section 6A. (a) Show that the set

V

is a vector space under natural operations of addition and scalar multiplication. (b) Show that there is a natural way to make

V

into a normed vector space and that with this norm,

V

is a Banach space. (c) Explain why (b) shows that every normed vector space is a subspace of some Banach space.