Linear extensions of relations between vector spaces

LetX and Y be vector spaces over the same field K. Following the terminology of Richard Arens [Pacific J. Math. 11 (1961), 9–23], a relation F of X into Y is called linear if λF (x) ⊂ F (λx) and F (x) + F (y) ⊂ F (x+ y) for all λ ∈ K \ {0} and x, y ∈ X. After improving and supplementing some former results on linear relations, we show that a relation Φ of a linearly independent subset E of X into Y can be extended to a linear relation F of X into Y if and only if there exists a linear subspace Z of Y such that Φ(e) ∈ Y |Z for all e ∈ E. Moreover, if E generates X, then this extension is unique. Furthermore, we also prove that if F is a linear relation of X into Y and Z is a linear subspace of X, then each linear selection relation Ψ of F |Z can be extended to a linear selection relation Φ of F . A particular case of this Hahn-Banach type theorem yields an easy proof of the existence of a linear selection function f of F such that f ◦F is also a function.