Linear Algebra Done Right · Problem 6C.13

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · riesz-representation

Problem

Suppose

\mathbf{F} = \mathbf{R}

and

V

is finite-dimensional. For each

v \in V

, let

\varphi_v

denote the linear functional on

V

defined by

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

for all

u \in V

. (a) Show that

v \mapsto \varphi_v

is an injective linear map from

V

V'

. (b) Use (a) and a dimension-counting argument to show that

v \mapsto \varphi_v

is an isomorphism from

V

onto

V'

. The purpose of this exercise is to give an alternative proof of the Riesz representation theorem (6.42 and 6.58) when

\mathbf{F} = \mathbf{R}

. Thus you should not use the Riesz representation theorem as a tool in your solution.