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Review of Weak Topology and Renormings

In this chapter, we discuss some basic tools from nonseparable Banach space theory that will be used in subsequent chapters. The first part concentrates on some fundamental results concerning Mackey and weak topologies. For example, the first section presents some of Grothendieck’s basic results on the dual Mackey topology on dual Banach spaces. The second section includes work of Odell, Rosenthal, Emmanuele, Valdivia, and others on the sequential agreement of dual Mackey and norm topologies in spaces that do not contain \(\ell _1\). In the third section, our attention turns to classical results of Dunford, Pettis, and Grothendieck on weak compactness in L1(μ) spaces and in the duals of C(K) spaces; this section ends with the Josefson-Nissenzweig theorem, which shows that, for all infinite-dimensional spaces X, SX* is weak*-sequentially dense in BX*. These results will be needed in Chapter 7.

Keywords

Banach Space Weak Topology Dual Pair Projectional Resolution Null Sequence 
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© Springer Science+Business Media, LLC 2008

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