TY - JOUR
T1 - Functionals on continuous functions
AU - Baxter, J. R.
AU - Chacon, R. V.
PY - 1974/4
Y1 - 1974/4
N2 - 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.
AB - 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.
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U2 - 10.2140/pjm.1974.51.355
DO - 10.2140/pjm.1974.51.355
M3 - Article
AN - SCOPUS:84972555293
SN - 0030-8730
VL - 51
SP - 355
EP - 362
JO - Pacific Journal of Mathematics
JF - Pacific Journal of Mathematics
IS - 2
ER -