By P. Wojtaszczyk

This is often an advent to trendy Banach area idea, during which functions to different parts comparable to harmonic research, functionality conception, orthogonal sequence, and approximation idea also are given prominence. the writer starts with a dialogue of vulnerable topologies, susceptible compactness, and isomorphisms of Banach areas sooner than continuing to the extra special research of specific areas. The ebook is meant for use with graduate classes in Banach house conception, so the must haves are a historical past in practical, advanced, and genuine research. because the in simple terms creation to the fashionable concept of Banach areas, it is going to be a necessary spouse for pro mathematicians operating within the topic, or to these drawn to employing it to different components of research.

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**Example text**

Proof: Let us fix a sequence (x;)�=l C X * such that i f x E V and x; (x) = 0 for n = 1, 2, . . then x = 0. g. take (vn)�= l dense in V and use the Hahn-Banach theorem to get x; E X* such that ll x; ll = 1 and x; (vn) = ll vn ll for n = 1, 2, 3, . . ) Using a standard diagonal argument we find a subsequence (ank )�1 such that limk -+oo x; (ank ) exists for every n = 1, 2, . . Let jj E X be any weak cluster point of the set {ank }� 1 C A; thus x; (y) = limk -+oo x; (ank ) for n = 1, 2, . .

Hilbert and M. Riesz. The general theory is presented in Banach [1932] . Banach, however, did not use the notion of weak topology, he relied exclusively on sequential arguments. This resulted in unnecessary separability assumptions in many theorems. The weak topology defined by weak neighbourhoods appears (for algebras of operators on Hilbert spaces) already in von Neumann [1930] . In the thirties it became clear that general topological notions are needed in order to establish non-separable theorems.

The implication ( c) =? ( a) for separable X is in Banach [1932] chapter XI Th. 13. It is interesting to note that he did not realize the converse, despite having a sequential version of Theorem 9. The equivalence ( a ) <=? (b) was known to Plessner before 1936 (see Lusternik [1936] ) . The equivalence ( a ) <=? ( c) was proved in Smulian[1939] , Bourbaki [1938] and Kakutani [1939] . As is clear from the above the content of this section is classical. The reader will find much more in Dunford-Schwartz [1958] .