Does every LCS has a convex balanced local base?

Question

Does every LCS--locally convex (topological vector) space has a convex balanced local base?

Then it implies every LCS is topologized with a countable number of separating seminorms. So there seems to exist a counterexample.

Answer 1

Yes, any locally convex space has a neighborhood basis around zero consisting of balanced convex sets (Wikipedia seems to take this as the definition but as far as I know including ``balanced" in the definition is non-standard, and in any case can be derived from the more general definition requiring the existence of a neighborhood basis around zero consisting of convex sets).

But this says nothing about first-countability. I'm not sure how you're drawing this conclusion. Not all locally convex spaces are metrizable (a first-countable, Hausdorff locally convex space is necessarily metrizable, and this is equivalent to its topology being defined by a countable set of seminorms whose common kernel is zero).

Answer 2

This is true. See corollary of theorem 1.14 in W. Rudin Functional analysis.

This property doesn't imply LCS have a countable separating family of seminorms.