Measure, Integration & Real Analysis · Problem 6D.4

L^p Spaces — Sheldon Axler

Mathematics — Measure Theory · finite-dimensional-linear-maps-continuous

Problem

Suppose

n \in \mathbf{Z}^+

and

V

is a normed vector space. Prove that every linear map from

\mathbf{F}^n

V

is continuous. [For this exercise,

\mathbf{F}^n

should be endowed with the norm

\|\cdot\|_\infty

as defined in Example 6.34.]