Linear Algebra Done Right · Problem 6A.25

Inner Product Spaces — Sheldon Axler

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

Problem

Suppose

p > 0

. Prove that there is an inner product on

\mathbf{R}^2

such that the associated norm is given by

\|(x, y)\| = (|x|^p + |y|^p)^{1/p}

for all

(x, y) \in \mathbf{R}^2

if and only if

p = 2