Measure, Integration & Real Analysis · Problem 6E.17

L^p Spaces — Sheldon Axler

Mathematics — Measure Theory · pointwise-limit-bounded-linear-maps

Problem

Suppose

V

is a Banach space,

W

is a normed vector space, and

T_1, T_2, \ldots

is a sequence of bounded linear maps from

V

W

such that

\lim_{k\to\infty} T_k f

exists for each

f \in V

. Define

T: V \to W

Tf = \lim_{k\to\infty} T_k f

for

f \in V

. Prove that

T

is a bounded linear map from

V

W

. [This result states that the pointwise limit of a sequence of bounded linear maps on a Banach space is a bounded linear map.]