In this article we introduce and prove properties of simplicial complexes in real linear spaces which are necessary to formulate Sperner's lemma. The lemma states that for a function f , which for an arbitrary vertex v of the barycentric subdivision B of simplex K assigns some vertex from a face of K which contains v, we can find a simplex S of B which satisfies f (S) = K (see [10]). The notati...

It is well-known that every weakly convergent sequence in ℓ 1 the norm topology (Schur's lemma). Phillips' lemma asserts even more strongly if a ( μ n ) ∈ N ∞ ′ converges pointwise on { 0 , } to 0, then its -projection 0. In this note we show how second category version of Schur's lemma, for which short proof included, can be used replace by any subsets contains all finite sets and having some ...