Remarks on certain function spaces

摘要: Section t . Introduction The norm topology in C(S) of uniform convergence on S (here is locally compact Hausdorff and denotes the algebra complex-valued bounded continuous functions S) an important one has been studied extensively, However, it does not seem appropriate to place if wishes study as a topological vector space or algebra, particularly relate properties with those S. In fact, when given this same C~S), where flS Stone-Cech compactification A more natural for non-compact would be which satisfies at least following properties: (1) multiplicative linear forms are (via evaluation) by points S, (2) coincides compact. Such was introduced Buck [3, 4], who called fl strict topology. Additional results (C(S), fl) have obtained recently CONWAY [7, 8, 9, 10], TODD [22], WELLS [24, 25], COLLINS [5], implement some degree notion that useful C(S). It purpose paper examine further fl), work being divided into four sections. section 3 shown always approximation property. These include theorem GROTHENDIECK [16, I, p. 185] but our method proof considerably direct revealing. 4 two theorems proved concerning special types approximate identities Banach Co(S ) complex vanish Do (supremum norm). known [4, 98] exist, here sequential exists only totally paracompact. 5 contains unrelated possibly interesting results: paracompact (DF) then M(S) a(M, C) sequentially complete Radon measures tr(M, weak * imposed C(S)), each uniformly real sequence upper bound (in particular, space) can contain no closed C* embedded copy N, numbers (and certainly must pseudocompact). final (section 6)