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4 the Weak-* Topology and the Banach-Alaoglu Theorem

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4 the Weak-* Topology and the Banach-Alaoglu Theorem 4 The weak-* topology and the Banach-Alaoglu Theorem 4.1 The weak-* topology Let E be a normed space over F. We know that E∗ is then a Banach space with its usual norm. However, sometimes the norm topology is too strong for our purposes, and we need to use a different topology, the weak-* topology . In particular, we will need the subspace topology induced by weak-* topology when we look at the Gelfand theory of commutative Banach algebras. 86 The weak-* topology on E∗ is an example of a topology on a vector space induced by a family of seminorms. Another example of this is the weak topology on E. (See books for more details.) Recall that FE is the set of all functions from E to F, and that this may be regarded as an infinite product of copies of F (assuming E =6 {0}). We may give this product the usual product topology. Note that E∗ is a subset of FE. 87 Definition 4.1 With notation as above, the weak-* topology on E∗ is the subspace topology on E∗ induced by the product topology on FE. It is the weakest topology on E∗ such that all of the functions f 7→ f(x) are continuous from E∗ to F (x ∈ E). This topology is denoted by σ(E∗, E ). Similarly, the weak topology on E, denoted by σ(E, E ∗), is the weakest topology on E such that all the functions x 7→ f(x) ( f ∈ E∗) are continuous, i.e., all the functionals in E∗ are continuous. 88 Remarks. (1) The algebraic dual E× is a closed subset of FE, but the topological dual E∗ need not be. (2) The closed unit ball of E∗ is a closed subset of FE. (3) Recall that the natural (or standard) embedding of E in its bidual E∗∗ is denoted by x 7→ xˆ. The image of E under this embedding is denoted b by E. We may give E∗∗ = ( E∗)∗ the weak-* topology σ(E∗∗ , E ∗) (as the dual space of E∗), and can b then give E the subspace topology, which we b call the relative weak-* topology on E. The standard embedding is then a ∗ b homeomorphism from (E, σ (E, E )) to E with the relative weak-* topology. 89 b (4) With notation as above, E is weak-* dense in E∗∗ . Also, the image of the closed unit ball of E is weak-* dense in the closed unit ball of E∗∗ . See books for the details, which are based on the sublinear functional version of the Hahn-Banach Theorem. 4.2 The Banach-Alaoglu Theorem We saw earlier that the closed unit ball of an infinite-dimensional normed space is never compact with the norm topology. However it may happen that this ball is compact with respect to one of our other topologies. 90 We begin with a lemma concerning compact subsets of FE where E is any non-empty set. This lemma may be regarded as a generalization of the Heine-Borel Theorem, and is based on Tychonoff’s Theorem. First recall that the coordinate projections on FE E are the maps px : F → F (x ∈ E) defined by E px(f) = f(x) ( f ∈ F ). The product topology on FE is the weakest topology on FE which makes all of the coordinate projections px continuous. We say that a subset S of FE is pointwise bounded if, for all x ∈ E, px(S) is a bounded subset of F. The bound may depend on x. 91 Lemma 4.2 With notation as above, a subset of FE is compact if and only if it is both closed in FE and pointwise bounded. We are now ready to prove the Banach-Alaoglu Theorem. Theorem 4.3 (Banach-Alaoglu) Let E be a normed space. Then the closed unit ball of E∗ is weak-* compact. The situation for the weak topology is rather different, as we see in the last result of this section. Theorem 4.4 Let E be a normed space. Then the closed unit ball of E is weakly compact if and only if E is a reflexive Banach space. 92.
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