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By A.M. Fink

ISBN-10: 3540067299

ISBN-13: 9783540067290

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Example text

6 that #[(Ao x A 1) U (A 1 x AO) U (A 1 x A 1)] = #A0. Set A = AOUAi and observe that (AxA)\(AoxA0) _ (AoxA1) U (A1 xAo) U (A1 xA1) because Ao and At are disjoint. Thus #[(AxA)\(AoxA0)] = #Ao = #A1, 20 1. Set-Theoretic Structures which ensures the existence of an injective function F1: A t -> X x X such that F1 (A 1) = (A x A)\ (A0 x A0). Now consider a function F from A to X x X defined as follows. F(x) _ FO(X) E Ao x Ao, X E A0, Fi(x) E (A x A) \ (Ao x A0), x E At = A \ Ao. F: A - X x X is injective (because Fo and F1 are injective functions with disjointranges) and F(A) = AxA (for F0(A0) U Ft(A1) = AxA).

Thus, according to Zorn's Lemma, ,7 contains a maximal function. Let FO be a maximal function of ,7 and let Ao be the domain of Fo, so that Fo(Ao) c X\Ao. Suppose #Ao < #(X\Ao) and show that, if X is an infinite set, then 34 1. ) and X\Ao is infinite - recall: the union of finite sets is finite. Now set A i = Ao U {xo} and consider the function Ft : A t - X defined by the formula Ft (x) = X E Ao, Fo(x) E Fo(Ao), X1 E X \ Fo(Ao), x = xo E X \ Ao. (c) Show that F1 E J. Since F0 = F, I& it follows that F0 < F1, which contradicts the fact that Fo is a maximal of 9 (for F0 A Fi ).

Example 2E. Let S be a nonempty set and let F be an arbitrary field. , the set of all scalar-valued functions on S, where "scalar-valued" stands for "F-valued"). Let vector addition and scalar multiplication be defined pointwise. That is, if x and y are functions in X and a is a scalar in F, then x + y and ax are functions in X defined by (x + y)(s) = x(s) + y(s) and (ax)(s) = a(x(s)) for every s E S. Now it is easy to show that X, when equipped with these two operations, in fact is a linear space over IF.

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Almost Periodic Differential Equations by A.M. Fink

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