Measure, Integration & Real Analysis · Problem 22

Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · hilbert-spaces-rank-one-operators

Problem

Prove that if

V

is a Hilbert space and

T: V \to V

is a bounded linear map such that the dimension of range

T

1

, then there exist

g, h \in V

such that

Tf = \langle f, g \rangle h

for all

f \in V