Although the weak topology of the unit ball is not metrizable in general, one can characterize weak compactness using sequences.
2.
In a Hilbert space, the weak compactness of the unit ball is very often used in the following way : every bounded sequence in has weakly convergent subsequences.
3.
Weak compactness of the unit ball provides a tool for finding solutions in reflexive spaces to certain convex continuous function on the unit ball of a reflexive space attains its minimum at some point in.
4.
A language " L " ?, ? is said to satisfy the weak compactness theorem if whenever ? is a set of sentences of cardinality at most ? and every subset with less than ? elements has a model, then ? has a model.
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