Measure, Integration & Real Analysis · Problem 18

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · bounded-linear-map-continuity-equivalences

Problem

Suppose

V

and

W

are normed vector spaces and

T: V \to W

is a linear map. Prove that the following are equivalent. (a)

T

is bounded. (b) There exists

f \in V

such that

T

is continuous at

f

. (c)

T

is uniformly continuous (which means that for every

\varepsilon > 0

, there exists

\delta > 0

such that

\|Tf - Tg\| < \varepsilon

for all

f, g \in V

with

\|f - g\| < \delta

). (d)

T^{-1}\big(B(0, r)\big)

is an open subset of

V

for some

r > 0