Schauder-Tychonoff fixed-point theorem ($T(K)\subseteq K⟹\exists x\in K:T(x)=x$)

Every continuous self-map of a nonempty compact convex subset of a locally convex space has a fixed point. Compactness is in the locally convex topology, which may be weaker than a norm topology. This version permits fixed-point arguments on a weak-star topology compact set of linear functionals. See Hartman's fixed-point chapter, Theorem 0.2.