Measure, Integration & Real Analysis · Problem 6E.12

L^p Spaces — Sheldon Axler

Mathematics — Measure Theory · bounded-below-iff-injective-closed-range

Problem

Suppose

T: V \to W

is a bounded linear map from a Banach space

V

to a Banach space

W

. Prove that

T

is bounded below if and only if

T

is injective and the range of

T

is a closed subspace of

W

. [A linear map

T: V \to W

from a normed vector space

V

to a normed vector space

W

is called bounded below if there exists

c \in (0, \infty)

such that

\|f\| \leq c\|Tf\|

for all

f \in V