Dual space of a complete vector space

Question

Let $(X, \mathcal{F}, \mathbb{P})$ be a Probability space. Consider the space of all functions with topology induced from convergence in Probability. I am interested in knowing the dual space of it. It's a well know result that the dual space of functions with uniform norm would be ba space. However, I did not find any result related to my question. Thanks for any help.

Answer 1

This is a classical example of a topological vector space without non-trivial continuous linear functionals. For the proof you show that for every $0$-neighbourhood $U$ its convex hull is the whole space (for $U=\lbrace f$ measurable$: |\phi(f)|<1\rbrace$ you then get $\phi=0$). Given a measurable function $f$ you just write it as $f(x)=\frac{1}{n} \sum_{k=1}^n nf(x) I_{A_k}(x)$ where $A_k$ form a partition of $X$ each having so small probability such that $nfI_{A_k} \in U$ (this requires some rather weak condition on the probability space).