Special Uniform Approximations of Continuous Vector-Valued Functions. Part I: Special Approximations in CX(T) Original Research Article

In this paper, we give special uniform approximations of functions u from the spaces CX(T) and C∞(T,X), with elements ū of the tensor products CΓ(T)⊗X, respectively C0(T,Γ)⊗X, for a topological space T and a Γ-locally convex space X. We call an approximation special, if ū satisfies additional constraints, namely supp v⊂u−1(X\{0}) and ū(T)⊂ co(u(T)) (resp. ⊂ co(u(T)∪{0})). In Section 3, we give three distinct applications, which are due exactly to these constraints: a density result with respect to the inductive limit topology, a Tietze–Dugundjiʹs type extension new theorem and a proof of Schauder–Tihonovʹs fixed point theorem.