2 problems

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The abundance of proper dense subspaces in infinite-dimensional topological vector spaces
Abundance conjecture. An infinite-dimensional TVS has infinitely many proper dense subspaces if and only if $w(X)\leq \dim(X)$ .
Conjectureopendense-subspaces functional-analysis topological-vector-spacesarXiv posted 2 days ago
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Existence of large dense subspaces with smooth norms in separable Banach spaces
Let $\mathcal{X}$ be a separable Banach space. A dense subspace $\mathcal{Y}$ of $\mathcal{X}$ has dimension continuum when its linear dimension is the cardinality of the continuum…
Conjectureopenbanach-spaces dense-subspaces smooth-normsarXiv posted 2 days ago

The abundance of proper dense subspaces in infinite-dimensional topological vector spaces