Measure, Integration & Real Analysis · Problem 10C.3

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · compact-operators-weak-convergence

Problem

Suppose

T

is a compact operator on a Hilbert space

V

and

f_1, f_2, \ldots

is a sequence in

V

such that

\lim_{n\to\infty}\langle f_n, g\rangle = 0

for every

g \in V

. Prove that

\lim_{n\to\infty}\|Tf_n\| = 0