If $X$ is an infinite-dimensional topological vector space which is the union of countably many finite-dimensional subspaces, prove that $X$ is of the first category in itself. Prove that therefore no infinitely-dimensional F-space has a countable Hamel basis
This is an exercise from Rudin book. But I can't construct a set of nowhere dense closed sets which have union equal $X$. Can anyone help me? Thanks.