Measure, Integration & Real Analysis · Problem 16

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · operator-norm-characterizations

Problem

Suppose

V

and

W

are normed vector spaces with

V \neq \{0\}

and

T: V \to W

is a linear map. (a) Show that

\|T\| = \sup\{\|Tf\| : f \in V \text{ and } \|f\| < 1\}

. (b) Show that

\|T\| = \sup\{\|Tf\| : f \in V \text{ and } \|f\| = 1\}

. (c) Show that

\|T\| = \inf\{c \in [0, \infty) : \|Tf\| \leq c\|f\| \text{ for all } f \in V\}

. (d) Show that

\|T\| = \sup\left\{\dfrac{\|Tf\|}{\|f\|} : f \in V \text{ and } f \neq 0\right\}