Continuous extension operators and convexity

Given a nonempty closed subset AA of a Hilbert space XX, we denote by L(A)L(A) the space of all bounded Lipschitz mappings from AA into XX, equipped with the supremum norm. We show that there is a continuous mapping Fc:L(A)↦L(X)Fc:L(A)↦L(X) such that for each g∈L(A)g∈L(A), Fc(g)|A=gFc(g)|A=g, Lip(Fc(g))=Lip(g), and Fc(g)(X)⊂clco(g(A)). We also prove that the corresponding set-valued extension operator is lower semicontinuous.