Linear operators on generalized 2-normed spaces (Q2762824)

One considers a generalization of the concept of 2-normed space (in the sense of S.~Gähler) by allowing 2-norms \(|\cdot,\cdot|\colon E\times E\rightarrow \mathbb{R}\) for which \(|x,y|=0\) out of the standard condition of linear independence of \(x\) and \(y\). A typical example is the inner space \(\mathbb{R}^n\), endowed with the 2-norm \(|x,y|=\|\langle x,y\rangle\|\). The main result asserts that the space of all 2-bounded linear operators from a generalized 2-normed space \((E_1,|\cdot,\cdot|_1)\) into a generalized symmetric sequentially complete 2-normed space \((E_2,|\cdot,\cdot|_2)\) is a Banach space.