Recent uses of connectedness in functional analysis
  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, connectedness 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. 
Biagio Ricceri