By Benz W.

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Be a family of transformations of a space S into a space T. , = T. ,. There is another method of introducing a topology into the family F of transformations. This method stems from the following consideration. Given a mapping f:S --t T, we know that the functional value f(x) is a continuous function of x. Can we so choose a topology for F that f(x) is a continuous function off? The answer is "yes," and the development which follows is largely due to R. H. Fox [78]. For each compact subset C of S and each open subset U ofT, let F(C, U) denote the collection of all mappings fin F, such that f(C) is contained in U.

Assuming that H contains an element other than 0, let h 1 be the element of H such that d(O, h 1 ) is a minimum. Then H consists of all multiples of h 1 . For if there were an element h 2 in H that was not a multiple of ht, then for some integer n, nh 1 - h 2 would be closer to zero than h 1 • Therefore His cyclic. 0 A product of topological groups is again a topological group. We utilize the usual direct product for the group operation and the Tychonoff topology. Precisely, if {G.. } is a collection of topological groups indexed by a set A = {a}, then the product IP AG..

We begin the discussion with several well-known examples that present some standard procedures. ExAMPLE 1. Let I dt:>note a closed interval [a, b] in El, and let C(l) be the collection of all real-valued continuous functions defined on I. We topologize C(I) by means of the following metric. For two functions f and g in C(l), define d(f, g) = maxr lf(x) - g(x)l. It is lrft as an rasy exercise for the rradcr to verify that this is indeed a metric. 1-11] 29 FUNCTION SPACES ExAMPLE 2. Let I (a, b] again, and let R(I) be the collection of all bounded real-valued functions on I, continuous or not.