Functionals on continuous functions

John Robert Baxter and Rafael Van Severen Chacon

Vol. 51 (1974), No. 2, 355–362

DOI: 10.2140/pjm.1974.51.355

Abstract

Let 𝒞(M) be the space of continuous functions on a compact metric space M. In a previous paper a class of nonlinear functionals Φ on 𝒞([0,1] × [0,1]) was constructed, such that each Φ satisfied:

1. lim_∥f∥→0Φ(f) = 0,
2. Φ(f + g) = Φ(f) + Φ(g) whenever fg = 0, and
3. Φ(f + α) = Φ(f) + Φ(α) for any constant α.

In this paper we show that the dimensionality of [0,1] × [0,1] is what makes the construction work. More precisely, we show that if Φ is a functional on 𝒞(M) satisfying (i), (ii), and (iii), and if the dimension of M is less than two, then Φ must be linear.

Mathematical Subject Classification 2000

Primary: 46E15

Secondary: 28A30

Milestones

Received: 19 January 1973

Published: 1 April 1974