Cardinal functions on TVS

Question

Its a well know fact that, in the class of metric spaces, the cardinal invariants density and weight agree. I would like to know if there is a example of a topological vector space $X$ for whose density $\operatorname{d}(X)$ does not agree with its weight $\operatorname{w}(X)$?

Answer 1

For a simple example, consider $\mathbb R^{\mathbb R}$ (i.e., the set of all functions $\mathbb R \to \mathbb R$) with the usual product topology (that is, the topology of pointwise convergence).

• As $\mathbb R$ is separable it follows by the Hewitt-Marczewski-Pondiczery Theorem that $\mathbb R^{\mathbb R}$ is separable, so $d ( \mathbb R^{\mathbb R} ) = \aleph_0$.

• As $\mathbb R$ is not countable (and the space $\mathbb R$ is not trivial), the product is not second-countable, so $w ( \mathbb R^{\mathbb R} ) > \aleph_0$.