On two conditional forms of the equation \(\| f(x+y)\|=\| f(x)+f(y)\|\)
Let \(X\) and \(E\) be real linear spaces and let \(E=(E,\|\circ\|)\) be a strictly normed space, that is, if \(u,v\in E\), \(u\neq 0\), \(v\neq 0\) and \(\| u+v\|=\| u\|+\| v\|\), then there exists a real \(t>0\) such that \(u=tv\). Let \(f\) be a function from \(X\) to \(E\). Without any regularity conditions on \(F\), the author shows that (a) \((x,y)\in X\times X\), \(f(x+y)\neq 0\Rightarrow\| f(x+y)\|=\| f(x)+f(y)\|\) (b) \((x,y)\in X\times X\), \(f(x)+f(y)\neq 0\Rightarrow\| f(x+y)\|=\| f(x)+f(y)\|\) (c) \(f(x+y)=f(x)+f(y)\) for all \((x,y)\in X\times X\) are equivalent.
scientific article; zbMATH DE number 202380