Measure, Integration & Real Analysis · Problem 19

Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · hilbert-spaces-orthogonal-projection

Problem

Suppose

U

and

W

are closed subspaces of a Hilbert space. Prove that

P_U P_W = 0

if and only if

\langle f, g \rangle = 0

for all

f \in U

and all

g \in W