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WEAK AND STRONG CONVERGENCE

DR GAGANDEEP

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THEOREM

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COROLLARY

If X is reflexive, then weak convergence and weak*convergence in X* are equivalent.

In case, X* is the dual of a non- reflexive space X, then weak convergence and weak*convergence are different. It is preferable to use weak* convergence in X* instead of weak convergence.

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THEOREM

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PROOF

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THEOREM

Let X be a separable Banach space and let M be a bounded subset of X*. Then, every sequence of bounded linear functional in X*

PROOF

Choose a countable dense subset A = { x1 , x2 , …}.

Let { fn} be a sequence in M. then

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THEOREM (BANACH-ALAOGLU)

Let X* be the dual of a Banach space X. then, the closed unit ball in X* is weak* compact.

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