The singular case in the stability of additive functions
A mapping \(\omega: Z\mapsto Y\) acting between quasi-normed spaces is quasi-additive if it satisfies an estimate \(\|\omega (x+y)-\omega (x)-\omega (y)\|_Y\leq\varepsilon (\|x\|_Z +\|y\|_Z)\) for some \(\varepsilon >0\) and all \(x,y\in Z\). This mapping \(\omega\) is said to be approximable if there is an additive map \(a: Z\mapsto Y\) such that \(\omega-a\) is continuous at the origin of \(Z\). Assuming that \(Z\) is a finite dimensional space and \(Y\) is a quasi-Banach space, it is proved that every quasi-additive map \(\omega: Z\mapsto Y\) is approximable. In the case when both \(Y\) and \(Z\) are possibly infinite dimensional quasi-Banach spaces it is proved that every quasi-additive map \(\omega: Z\mapsto Y\) is approximable if and only if every homogeneous quasi-additive map \(\omega: Z\mapsto Y\) is asymptotically linear.
scientific article; zbMATH DE number 1763472
scientific article