2-functionals and some extension theorems in linear spaces
Given linear subspaces M, L of a real linear space E, the authors define on \(M\times L\) the concept of a linear 2-functional f and a convex 2- functional p. These can easily be shown to be equivalent to f being a bilinear functional and p being a biseminorm (i.e., \(p(\cdot,y)\) and \(p(x,\cdot)\) are seminorms on M and L, respectively, for fixed \(y\in L\), \(x\in M)\). When L is finite dimensional with a basis \(\{e_ i\}^ n_{i=1}\) and \(p(x,\sum \alpha_ ie_ i)=\sum p(x,\alpha_ ie_ i),\) the authors prove that a linear 2-functional f dominated by p on \(M\times L\) can be extended to a linear 2-functional F dominated by p on \(E\times L\) (here p has domain \(E\times L)\). The proof can be simplified using the comments above. The Hahn-Banach theorem can be applied to each of \(f(\cdot,e_ i)\) to produce extensions \(F(\cdot,e_ i)\) on E, \(i=1,2,...,n\). Then F can be defined on \(E\times L\) in the obvious manner by means of the n extensions so obtained.
scientific article; zbMATH DE number 3855738