On the uniform density of C(X) C(Y) in C(X × Y)

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• 作者：P.D. Allenby ; C.C.A. Labuschagne
• 关键词：s ; Uniform convergence ; Space of continuous functions
• 刊名：Indagationes Mathematicae
• 出版年：2009
• 出版时间：March 2009
• 年：2009
• 卷：20
• 期：1
• 页码：19-22
• 全文大小：78 K

文摘

We prove that if X and Y are compact Hausdorff spaces, then every f C(X × Y)₊, i.e. f(x, y) ≥ 0 for all (x, y) X × Y, can be approximated uniformly from below and above by elements of the form , where f_i C(X)₊ and g_i C(Y)₊ for i = 1, 2, …, n. The proof uses only elementary topology. We use this result, in conjuction with Kakutani's M-spaces representation theorem, to obtain an alternative proof for a known property of Fremlin's Riesz space tensor product of Archimedean Riesz spaces.