## 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)].

### MSC:

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

### Citations:

Zbl 0145.389; Zbl 0165.499