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FUNCTIONAL ANALYSIS LECTURE NOTES: WEAK AND WEAK* …

FUNCTIONAL ANALYSIS LECTURE NOTES: WEAK AND WEAK* CONVERGENCECHRISTOPHER and Weak* convergence of VectorsDefinition a normed linear space, and letxn,x We say thatxnconverges,converges strongly, orconverges in normtox, and writexn x, iflimn kx xnk= We say thatxnconverges weaklytox, and writexnw x, if X ,limn hxn, i=hx, Show that strong convergence implies weak Show that weak convergence does not imply strong convergence in general (look for aHilbert space counterexample).If our space is itself the dual space of another space, then there is an additional mode ofconvergence that we can consider, as a normed linear space, and suppose that n, X.

FUNCTIONAL ANALYSIS LECTURE NOTES: WEAK AND WEAK* CONVERGENCE CHRISTOPHER HEIL 1. Weak and Weak* Convergence of Vectors Definition 1.1. Let …

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