4 Recent uses of connectedness in functional analysis

Recent uses of connectedness in functional analysis BIAGIO RICCERI Perhaps, it is not too far from the truth to say that, among the great concepts (as compactness, completeness, order, convexity) on which functional analysis is based, con-nectedness is relatively less popular, though this does not mean that it is less useful than the others. The aim of this lecture is just to support this latter sentence, focusing some recent results where connectedness plays a central role. Our starting point is Theorem 1 below. Before stating it, to give the reader the convenience to realize analogies and differences, we recall, grouped together in Theorem A, three very famous results due to K. A x = {v ∈ Y : (x, v) ∈ A} and A y = {u ∈ X : (u, y) ∈ A}. F be two real Hausdorff locally convex topological vector spaces, let X ⊆ E, Y ⊆ F be two non-empty compact convex sets, and let S, T be two subsets of X × Y. Assume that at least one of the following three sets of conditions is satisfied: (α) S y is convex for each y ∈ Y , S x is open in Y for each x ∈ X, T x convex for each x ∈ X, and T y is open in X for each y ∈ Y ; (β) S, T are closed, S y is convex for each y ∈ Y , and T x is convex for each x ∈ X; (γ) S y is convex for each y ∈ Y , S x is open in Y for each x ∈ X, T is closed, and T x is convex for each x ∈ X. Then, at least one of the following assertions does hold: (a) p X (T) = X. (b) p Y (S) = Y. (c) S ∩ T = ∅. In [18], we pointed out that, when Y is a segment, Theorem A is still true assuming simply that the sections S y are connected. More precisely, we have the following THEOREM 1 ([18], Theorem 2.3).-Let X, Y be two topological spaces, with Y admitting a continuous bijection onto [0, 1], and let S, T be two subsets of X × Y , with S connected and, for each x ∈ X, T x connected. Moreover, assume that either T y is open for each y ∈ …