Linear Algebra Done Right · Problem 6A.23

Inner Product Spaces — Sheldon Axler

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

Problem

Suppose

v_1, \ldots, v_m \in V

are such that

\|v_k\| \leq 1

for each

k = 1, \ldots, m

. Show that there exist

a_1, \ldots, a_m \in \{1, -1\}

such that

\|a_1 v_1 + \cdots + a_m v_m\| \leq \sqrt{m}.