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  • Print publication year: 2020
  • Online publication date: March 2020

Appendix C - The Banach–Alaoglu Theorem

from APPENDICES

Summary

We define weak convergence and give some examples, including a proof of Schur’s Theorem that weak and strong convergence coincide in l^1. We also show that closed convex subsets of Banach spaces are weakly closed. We then introduce weak-* convergence and prove two powerful weak compactness theorems: Helly’s Theorem for weak-* convergence in the duals of separable Banach spaces and a weak sequential compactness theorem in reflexive Banach spaces.