Banach–Alaoglu Theorem
The closed dual unit ball is compact for pointwise evaluation on the original normed space. The separate sequential theorem identifies where separability enters arguments that extract convergent subsequences.
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normed vector space · continuous dual space · Topological Compactness
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Start here: 245B, Notes 11: The strong and weak topologies
Terence Tao · Lecture notes · English · Free full text
§3 The weak and weak* topologies, Theorem 3 and Theorem 4 (Sequential Banach-Alaoglu theorem)
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Functional Analysis in the library · Preparation and skip guidance