Weak compactness refers to a property of sets in a topological vector space where every net that converges weakly has a convergent subnet whose limit is contained in the set. This concept is crucial when discussing weak topologies, as it ensures that certain convex sets remain 'nice' and manageable under weak convergence. Weak compactness plays an essential role in functional analysis and convex geometry, particularly in establishing the continuity of linear functionals and understanding dual spaces.
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