Measure, Integration & Real Analysis · Problem 15

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · quotient-normed-spaces

Problem

For readers familiar with the quotient of a vector space and a subspace: Suppose

V

is a normed vector space and

U

is a subspace of

V

. Define

\|\cdot\|

V/U

\|f + U\| = \inf\{\|f + g\| : g \in U\}.

(a) Prove that

\|\cdot\|

is a norm on

V/U

if and only if

U

is a closed subspace of

V

. (b) Prove that if

V

is a Banach space and

U

is a closed subspace of

V

, then

V/U

(with the norm defined above) is a Banach space. (c) Prove that if

U

is a Banach space (with the norm it inherits from

V

) and

V/U

is a Banach space (with the norm defined above), then

V

is a Banach space.