Linear Algebra Done Right · Problem 6A.28

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · inner-products-and-norms

Problem

A norm on a vector space

U

is a function

\|\cdot\|: U \to [0, \infty)

such that

\|u\| = 0

if and only if

u = 0

\|\alpha u\| = |\alpha|\|u\|

for all

\alpha \in \mathbf{F}

and all

u \in U

, and

\|u + v\| \leq \|u\| + \|v\|

for all

u, v \in U

. Prove that a norm satisfying the parallelogram equality comes from an inner product (in other words, show that if

\|\cdot\|

is a norm on

U

satisfying the parallelogram equality, then there is an inner product

\langle \cdot, \cdot \rangle

U

such that

\|u\| = \langle u, u \rangle^{1/2}

for all

u \in U