I am looking for fast ways to ensure that a given set is a subset of topologically locally convex space.
I have already read the posts post1:seminorms-in-locally-convex-spaces, locally-convex-space-via-seminorms, post2:topological-vector-space-locally-convex and also wikipage. However I am still not sure if semi-norm is only fast way to ensure the locally convexity and how can we ensure that a space (or subset of a topological vector space) has semi-norm or not!
For example I am not sure that if the following convex sets has semi-norm
- Probably space over $\mathbb{R}^2$
- The subspace of continuous probability density function over the above probability measure space
Topological vector space is locally convex iff its topology is generated by some family of seminorms. When you ask your question you implicitly assume that topology of your topological vector space is already given. But wait, for most of topological vector spaces topology is described only for the neighbourhood of zero and often this description is made in terms of seminorms
As indicated above by "probability space over $\mathbb{R}^2$" you mean the space of probability measures on $\mathbb{R}^2$. This is convex set but not a linear space (because it does npt contain $0$ measure). Since this set is not linear space, there is no point to ask whether it is locally convex topological space, because the very definition of locally convex topological space assumes that we are already given some linear space and the word locally convex here means that topology given this linear space is "nice" in certain sense.