Uniform convergence for convexification of dominated pointwise convergent continuous functions

Abstract

The Lebesgue dominated convergence theorem of the measure theory implies that the Riemann integral of a bounded sequence of continuous functions over the interval [ 0,1] pointwise converging to zero, also converges to zero. The validity of this result is independent of measure theory, on the other hand, this result together with only elementary functional analysis, can generate measure theory itself. The mentioned result was also known before the appearance of measure theory, but the original proof was very complicated. For this reason this result, when presented in teaching, is generally obtained based on measure theory. Later, Eberlein gave an elementary, but still relatively complicated proof, and there were other simpler proofs but burdened with complicated concepts, like measure theory. In this paper we give a short and elementary proof even for the following strenghened form of the mentioned result: a bounded sequence of continuous functions defined on a compact topological space K pointwise converging to zero, has a suitable convexification converging also uniformly to zero on $K,$ thus, e.g., the original sequence converges weakly to zero in C(K). This fact can also be used in the proof of the Krein-Smulian theorem. The usual proof beyond the simple tools of the functional analysis, uses heavy embedding theorems and the Riesz' representation theorem with the whole apparatus of measure theory. Our main result, however, reduces the cited proof to a form in which we need abstract tools only, namely the Hahn-Banach separation theorem and Alaoglu's theorem, without Riesz' representation or any statement of measure theory.