Orthogonally additive functionals on sequence spaces. (English) Zbl 0613.46007

A functional F defined on a sequence space is said to be orthogonally additive if \(F(x+y)=F(x)+F(y)\) whenever \(x_ ky_ k=0\) for every k, where \(x=\{x_ k\}\) and \(y=\{y_ k\}\). A sequence space is solid if \(x\in X\) whenever \(| x| \leq | y|\) for some \(y\in X\). The authors proved a representation theorem for orthogonally additive functionals on solid sequence spaces and, in particular, on \(\ell_ p\), where \(1\leq p<\infty\) and \(c_ 0\). For a function version, see, for example, [N. Friedman and M. Katz, Can. J. Math. 18, 1264- 1271 (1966; Zbl 0145.389); V. J. Mizel and K. Sundaresan, Arch. Ration. Mech. Anal. 30, 102-126 (1968; Zbl 0165.499)].


46A40 Ordered topological linear spaces, vector lattices
47B60 Linear operators on ordered spaces
46A45 Sequence spaces (including Köthe sequence spaces)