A generalization of Hahn--Banach extension theorem
Let \(X\) and \(Y\) be real linear topological spaces with \(Y\) being partially ordered by a closed pointed convex cone \(K\), and let \(C\subset X\) be a convex set. A set-valued map \(F\colon\,C\to2^Y\) is said to be \(K\)-convex if, for every \(x,y\in C\) and \(\lambda\in(0,1)\), one has \(\lambda F(x)+(1-\lambda)F(y)\subset F\bigl(\lambda x+(1-\lambda) y\bigr)+K\). The authors define a set-valued map \(H\colon\,X\to2^Y\) to be affinelike if there exist a continuous linear map \(f\colon\,X\to Y\) and a nonempty convex set \(M\subset Y\) such that \(H(x)=f(x)+M\) for all \(x\in X\). The following ``Hahn--Banach theorem'' is obtained. Suppose that \((Y,K)\) is order-complete, and let a \(K\)-convex map \(F\colon\,C\to 2^Y\) be such that \(F(x)\not=\varnothing\) for all \(x\in C\). Then, whenever \(X_0\subset X\) is a proper subspace such that \(X_0\cap{\text{Cor}}\,C\not=\varnothing\) and \(H\colon\,X_0\to2^Y\) is an affinelike map such that \(F(x)-H(x)\subset K\) for all \(x\in X_0\cap C\), there exists an affinelike map \(L\colon\,X\to2^Y\) such that \(L(x)=H(x)\) for all \(x\in X_0\cap C\) and \(F(x)-L(x)\subset K\) for all \(x\in C\). (Here, \(\text{Cor}\,C\) is the algebraic interior of \(C\), i.e., the set of the points \(x\in C\) with the following property: for all \(z\in X\), there exists a \(\delta>0\) such that, for all \(\lambda\in(0,\delta)\), one has \(x+\lambda z\in C\).) Several corollaries are discussed.
scientific article; zbMATH DE number 2136658
scientific article