Measure, Integration & Real Analysis · Problem 6E.13

L^p Spaces — Sheldon Axler

Mathematics — Measure Theory · bounded-inverse-theorem-w-not-banach

Problem

Give an example of a Banach space

V

, a normed vector space

W

, and a one-to-one bounded linear map

T

V

onto

W

such that

T^{-1}

is not a bounded linear map of

W

onto

V

. [This exercise shows that the hypothesis in the Bounded Inverse Theorem (6.83) that

W

is a Banach space cannot be relaxed to the hypothesis that

W

is a normed vector space.]