Strong Convergence and Weak Convergence
In this chapter, we shall be concerned with certain basic facts pertaining to strong-, weak- and weak* convergences, including the comparison of the strong notion with the weak notion, e.g., strong- and weak measurability, and strong- and weak analyticity
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Conversely, it is easy to see that any "p satisfying (6), (7) and (8) defines an IE LOO(5,~, m)' through (10) and that (11) is true. Therefore, we have proved that LOO(5,~, m)' is the space of all set Iunctions e satisfying (6), (7) and (8) and normed by the right hand side of (11), the so-called total variation of "P. Remark. We have so far proved that U(5, ~, m) is reflexive when 1 < p < 00 . However, the space V(5, ~ , m) is, in general, not reflexive.
Example 6. C (5)' .
Let 5 be a compact topological space. Then the dual space C (5)' of the space C (5) of complex-valued continuous functions on 5 is given as follows. To any 1E C (5)' , there corresponds a uniquely determined complex Baire measure p, on 5 such that
I(x) = !x(s)p,(ds) whenever xEC(S), s
and hence 11/11 =
sup
sup!",(s)(;S;l s
11 (s) x
I
fl (ds) = the total variation of fl on S.
(12)
(13)
Conversely, any Baire measure p, on 5 such that the right side of (13) is finite , defines a continuous linear functional IE C(5)' through (12) and we have (13). Moreover , if we are concerned with a real functional I on areal B-space C(5), then the corresponding measure u is real-valued ; if, moreover 1 is positive, in the sense that 1(x) > 0 for non-negative functions x (s), then the corresponding measure p, is positive, i.e.,p, (B) > 0 for every B E ~ . Remark. The result stated above is known as the F. Riesz-A. MarkovS. Kakutani theorem, and is one of the fundamental theorems in topological measures. For the proof, the reader is referred to standard text books on measure theory, e.g., P. R. HALMOS [1] and N. DUNFORD]. SCHWARTZ [1J.
References for Chapter IV For the Hahn-Banach theorems and related topics , see BANACH [L], BOURBAKI [2J and KOTHE [1]. It was MAZUR [2J who noticed the importance of convex sets in normed linear spaces. The proof of Reily's theorem given in this book is due to Y. MIMURA (unpublished).
v.
Strong Convergence and Weak Convergence
In this chapter, we shall be concerned with certain basic facts pertaining to strong-, weak- and weak* convergences, including the comparison of the strong notion with the weak notion, e.g., strong- and weak measurability, and strong- and weak analyticity. We also discuss the
K. Yosida, Functional Analysis © Springer-Verlag Berlin Heidelberg 1965
120
V. Strong Convergence and Weak Convergence
integration of B-space-valued functions, that is, the theory of Bochner's integrals. The general theory of weak topologies and duality in locally convex linear topological spaces will be given in the Appendix. 1. The Weak Convergence and The Weak* Convergence Weak Convergence Definition 1. A sequence {x,,} in a normed linear space X is said to be weakly convergent if a finite lim / (x,,) exists for each / E X;; {x,,} is n->OO
said to converge weakly to an element xooE X if lim /(x,,) = /(xoo) for n-+OO
X;.
aU I E In the latter case, X oo is uniquely determined, in virtue of the Hahn-Banach theorem (Corollary 2 of Theorem 1 in Chapter IV, 6);we shall write w-lim x" = Xoo or, in short, x" -+ X
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