## On the space $$\omega$$.(English)Zbl 0635.46007

The author solves the following two problems concerning the space $$\omega$$ of all real sequences.
1. Given a real sequence space X, give necessary and sufficient conditions on the infinite matrix A, such that A defines a matrix transformation from $$\omega$$ to X.
2. Characterize the orthogonally additive functionals on $$\omega$$.
A functional $$f: \omega\to R$$ is orthogonally additive if for $$x=(x_ k)\in \omega$$ and $$y=(y_ k)\in \omega$$ with $$x_ ky_ k=0$$, for all k one has $$f(x+y)=f(x)+f(y).$$
Theorem 1: Solution of 1.
A: $$\omega\to X\Leftrightarrow$$ A has finitely many nonzero columns and these are the elements of X.
Theorem 2: Solution of 2.
f is orthogonally additive $$\Leftrightarrow$$ $$f(x)=\sum^{\infty}_{k=1}g(k,x)$$ with
i) $$g(k,0)=0$$, for all k
ii) g(k,.) is continuous on R
iii) There are $$L>0$$ and $$(a_ k)$$ with $$\sum_{k}a_ k$$ convergent such that: $$| g(k,t)| \leq a_ k$$ for all t and $$k\geq L.$$
This last result is a special case of theorem 2 in “Characterization of orthogonally additive operators on sequence spaces” by Chew Tuan Seng [South Est Asian Bull. Math. 11, 39-44 (1987; review below)].

### MSC:

 46A45 Sequence spaces (including Köthe sequence spaces) 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

### Keywords:

space of all real sequences; matrix transformation

Zbl 0635.46008