Linear Algebra Done Right · Problem 6C.8

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · orthogonal-projections

Problem

Suppose

U

is a finite-dimensional subspace of

V

and

v \in V

. Define a linear functional

\varphi \colon U \to \mathbf{F}

\varphi(u) = \langle u, v \rangle

for all

u \in U

. By the Riesz representation theorem (6.42) as applied to the inner product space

U

, there exists a unique vector

w \in U

such that

\varphi(u) = \langle u, w \rangle

for all

u \in U

. Show that

w = P_U v