Measure, Integration & Real Analysis · Problem 6E.9
L^p Spaces — Sheldon Axler
Mathematics — Measure Theory · infinite-dimensional-banach-no-countable-basis
Problem
Prove that there does not exist an infinite-dimensional Banach space with a countable basis.
[This exercise implies, for example, that there is not a norm that makes the vector space of polynomials with coefficients in F into a Banach space.]