Measure, Integration & Real Analysis · Problem 6D.5

L^p Spaces — Sheldon Axler

Mathematics — Measure Theory · inverse-bounded-linear-map

Problem

Suppose

n \in \mathbf{Z}^+

V

is a normed vector space, and

T: \mathbf{F}^n \to V

is a linear map that is one-to-one and onto

V

. (a) Show that

\inf\{\|Tx\| : x \in \mathbf{F}^n \text{ and } \|x\|_\infty = 1\} > 0.

(b) Prove that

T^{-1}: V \to \mathbf{F}^n

is a bounded linear map. [For this exercise,

\mathbf{F}^n

should be endowed with the norm

\|\cdot\|_\infty

as defined in Example 6.34.]