Please verify this definition of locally convex vector space

Question

A locally convex vector space is a vector space $V$ equipped with a family $P$ of separating semi-norms. Is this a correct definiton?

So to determine whether a norm induces a locally convex topology on a vector space is it enough to show that:

• $P$ is a family of semi-norms on $V$

• $P$ if separating $V$

Answer 1

The standard definition of a locally convex space is

A locally convex space is a topological vector space $E$ such that the convex neighbourhoods form a neighbourhood basis at each point.

By the translation-invariance of vector space topologies, one can alternatively only demand that the convex neighbourhoods of $0$ form a neighbourhood basis of $0$.

It is a fact that a topological vector space is locally convex if and only if its topology can be induced by a family of seminorms.

If your definition of topological vector spaces requires them to be Hausdorff, then the family of seminorms must be separating.

Thus we can use an alternative but equivalent definition of a locally convex space:

A locally convex space is a vector space endowed with a topology induced by a (separating) family of seminorms.

If we just say that the space is equipped with a (separating) family of seminorms, we have not explicitly fixed a topology on the space, so I would advise against using that as a definition.

A norm on a space gives rise to a separating singleton-family of seminorms, so induces a Hausdorff locally convex topology.

Conversely, a topological vector space is normable (its topology can be induced by a norm) if and only if it is locally convex and locally bounded (there is a bounded neighbourhood of $0$).