I believe the question is self-explanatory. After browsing in the web the following definitions stand out:
Assume we have a dual pair $(X,X')$
- The topology of uniform convergence on the convex balanced subsets of $X'$ that are compact in the weak topology $\sigma(X,X')$. (from Encyclopedia of Mathematics)
- Is a polar topology defined on $X$ by using the set of all absolutely convex and weakly compact sets in $X'$. (from Wikipedia)
However, neither can be considered ''formal'' definition and for an amateur in functional analysis the definitions are very obscure.
Wikipedia's definition seem to be more suitable for a layperson, but it does not specify how to ''use'' the absolutely compact sets and the weakly compact sets.
Furthermore, I would greatly appreciate some intuition regarding the relevance of the concept and some solid references to learn about the subject. I came across this concept while researching social choice [Shinotsuka, Tomoichi. "Equity, continuity, and myopia: a generalization of Diamond’s impossibility theorem." Social Choice and Welfare 15.1 (1997): 21-30].