Local base of normed spaces

Question

Here is my question:

Does every normed space has a countable local base?

Can you give me a hint to prove or disprove it?

Is it true to say every normed space is topological vector space and TVS has a countable local base if and only if it is metrizable?

Answer 1

A normed space is metrisable (metric $d(x,y)=\|x-y\|$) and so first countable, using the balls of radius $\frac{1}{n}$ around each point as a countable local base.

It is true that a TVS is metrisable iff it has a countable local base at $0$ (we only need the group structure, in fact), but this in general quite hard (Birkhoff-Kakutani theorem). This is the trivial direction of that.