Measure, Integration & Real Analysis · Problem 6E.10
L^p Spaces — Sheldon Axler
Mathematics — Measure Theory · open-mapping-theorem-banach-hypothesis
Problem
Give an example of a Banach space V, a normed vector space W, a bounded linear map T of V onto W, and an open subset G of V such that T(G) is not an open subset of W.
[This exercise shows that the hypothesis in the Open Mapping Theorem that W is a Banach space cannot be relaxed to the hypothesis that W is a normed vector space.]