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Quasi-additive mappings. (English) Zbl 1059.46003
A map $f$ between Abelian topological groups is called quasi-additive if the function of two variables $f\left(x+y\right)-f\left(x\right)-f\left(y\right)$ is continuous at the origin. Obvious examples are additive maps, maps which are continuous at the origin, and sums of such maps. The author proves that there are no others, in case the range is a quasi-Banach space, and the domain is a vector space equipped with a Hausdorff weak topology. This unifies several known results, with a proof which is no more technical than those existing in the literature. An important special case is when the domain is the real line. The same conclusion also holds if the range is a quasi-Banach space, and the domain is the space of measurable functions on a finite measure space, equipped with the topology of convergence in measure. An application is that, within the category of Abelian topological groups, any extension of a quasi-Banach space by a quasi-Banach space is again a quasi-Banach space. It remains open whether these results can be generalized from quasi-Banach spaces to $F$-spaces.
##### MSC:
 46A16 Non-locally convex linear spaces 39B82 Stability, separation, extension, and related topics 47H99 Nonlinear operators