Measure, Integration & Real Analysis · Problem 6E.11

L^p Spaces — Sheldon Axler

Mathematics — Measure Theory · open-mapping-theorem-normed-hypothesis

Problem

Show that there exists a normed vector space

V

, a Banach space

W

, a bounded linear map

T

V

onto

W

, and an open subset

G

V

such that

T(G)

is not an open subset of

W

. [This exercise shows that the hypothesis in the Open Mapping Theorem that

V

is a Banach space cannot be relaxed to the hypothesis that

V

is a normed vector space.]