For a given undirected graph with n vertices the paper defines four norms on , namely where (resp. ) stands for the family of all maximal cliques in G (resp. , the complement of G). The goal of this note is to demonstrate the usefulness of some notions and techniques from functional analysis >l11<in graph theory by showing how Theorem 2.1 (G is -perfect if and only if the norms are equal) together with >l10<the reflexivity of the space equipped with either of the norms above easily yield one new result (Theorem 2.2) and two known characterizations of perfect graphs (Theorems 2.3-2.4). Namely, Theorem 2.2 provides a characterization of -perfection that is strongly related to that of Lovász. It implies that the Lovász inequality is exactly the classical Schwartz inequality for the space, restricted to (0, 1) vectors x, y satisfying x=y. Theorem 2.3 is well known as the Perfect Graph Theorem, while Theorem 2.4, due to V. Chvátal and D.R. Fulkerson, characterizes -perfection of a graph G in terms of the equality between the vertex packing polytope of G and the fractional vertex packing polytope of G.
