Weak Convergence (Hilbert space)
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High Quality Content by WIKIPEDIA articles! Since every closed and bounded set is weakly relatively compact (its closure in the weak topology is compact), every bounded sequence xn in a Hilbert space H contains a weakly convergent subsequence. Note that closed and bounded sets are not in general weakly compact in Hilbert spaces (consider the set consisting of an orthonormal basis in an infinitely dimensional Hilbert space which is closed and bounded but not weakly compact since it doesn't contain 0). However, bounded and weakly closed sets are weakly compact so as a consequence every convex bounded closed set is weakly compact.…mehr

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High Quality Content by WIKIPEDIA articles! Since every closed and bounded set is weakly relatively compact (its closure in the weak topology is compact), every bounded sequence xn in a Hilbert space H contains a weakly convergent subsequence. Note that closed and bounded sets are not in general weakly compact in Hilbert spaces (consider the set consisting of an orthonormal basis in an infinitely dimensional Hilbert space which is closed and bounded but not weakly compact since it doesn't contain 0). However, bounded and weakly closed sets are weakly compact so as a consequence every convex bounded closed set is weakly compact.