Linear Algebra Done Right · Problem 16

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · inner-products

Problem

Suppose

V

is finite-dimensional. Suppose

\langle \cdot, \cdot \rangle_1

\langle \cdot, \cdot \rangle_2

are inner products on

V

with corresponding norms

\|\cdot\|_1

and

\|\cdot\|_2

. Prove that there exists a positive number

c

such that

\|v\|_1 \leq c\|v\|_2

for every

v \in V