Functionals on continuous functions

J. R. Baxter, R. V. Chacon

Research output: Contribution to journalArticlepeer-review

Abstract

Let e(M) be the space of continuous functions on a compact metric space M. In a previous paper a class of nonlinear functionals Φ on e([0, 1] x [0, 1]) was constructed, such that each 0 satisfied: (i) lim(FORMULA PRESENTED) (ii) (FORMULA PRESENTED) (iii) (FORMULA PRESENTED) for any constant α. In this paper we show that the dimensionality of [0, 1] x [0, 1] is what makes the construction work. More precisely, we show that if Φ is a functional on e(M) satisfying(i),(ii), and (iii), and ifthe dimension of M is less than two, then Φ must be linear.

Original languageEnglish (US)
Pages (from-to)355-362
Number of pages8
JournalPacific Journal of Mathematics
Volume51
Issue number2
DOIs
StatePublished - Apr 1974

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