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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 $f(x+y)-f(x)-f(y)$ 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.

46A16Non-locally convex linear spaces
39B82Stability, separation, extension, and related topics
47H99Nonlinear operators
Full Text: DOI
[1] Sánchez, F. Cabello: The singular case in the stability of additive functions. J. math. Anal. appl. 268, 498-516 (2002) · Zbl 1006.46002
[2] F. Cabello Sánchez, Quasi-homomorphisms, Fund. Math., submitted for publication
[3] Castillo, J. M. F.; González, M.: Three-space problems in Banach space theory. Lecture notes in math. 1667 (1997) · Zbl 0914.46015
[4] Domański, P.: On the splitting of twisted sums, and the three space problem for local convexity. Studia math. 82, 155-189 (1985) · Zbl 0582.46004
[5] Hyers, D. H.: On the stability of the linear functional equation. Proc. nat. Acad. sci. USA 271, 222-224 (1941) · Zbl 0061.26403
[6] Hyers, D. H.; Isac, G.; Rassias, Th.M.: Stability of functional equations in several variables. (1998) · Zbl 0907.39025
[7] Kalton, N.: The three space problem for locally bounded F-spaces. Compositio math. 37, 243-276 (1978) · Zbl 0395.46003
[8] Kalton, N.; Peck, N. T.: Quotients of $Lp(0,1)$ for 0\leqslantp1. Studia math. 64, 65-75 (1979) · Zbl 0393.46007
[9] Kalton, N.; Peck, N. T.; Roberts, W.: An F-space sampler. London math. Soc. lecture note ser. 89 (1984) · Zbl 0556.46002
[10] Megginson, R. E.: An introduction to Banach space theory. Graduate texts in math. (1998) · Zbl 0910.46008
[11] Roelcke, W.; Dierolf, S.: On the three-space-problem for topological vector spaces. Collect. math. 32, 13-35 (1981) · Zbl 0489.46002
[12] Rolewicz, S.: Metric linear spaces. (1984) · Zbl 0526.49018
[13] Skof, F.: Sull’approssimazione delle applicazioni localemente ${\delta}$-additive (On the approximation of locally ${\delta}$-additive mappings). Atti accad. Sci. Torino cl. Sci. fis. Mat. natur. 117, 377-389 (1983) · Zbl 0794.39008