imization of functionals, Markov-Kakutani fixed point theorem, Hahn- Banach theorem. 2010 AMS Subject Classification: Primary: 52A07, 32F32, 32F27, Secondary: 47H04, 47H10, 47N10 The aim of this expository paper is to show that a number of landmark results of nonlinear functional analysis can be quickly obtained from a particular version of the KKM principle at little cost. This ”elementary KKM principle” is due to A. Granas and M. Lassonde in the framework of super-reflexive Banach spaces [10]. It is extended to arbitrary topological vector spaces, under a more general compactness hypothesis, with a simpler proof based on the separation of closed convex subsets in a Euclidean space (a result usually discussed in a first course of continuous optimization) and an intersection theorem of V. L. Klee [17]. A similar approach is followed to formulate a coincidence theorem for so-called von Neumann relations . The methods outlined here allow for a shorter and simpler alternative treat- ment of existence results of functional analysis that avoids involved and deeper principles that require sophistication and investment in time. The KKM princi- ple is a striking example of such fundamental results. Indeed, using the Sperner lemma as a starting point, three of the greatest topologists of all times, Pol- ish academician S. Mazurkiewicz and two of his former doctoral students, B. Knaster and K. Kuratowski published in 1929 the celebrated KKM lemma: a remarkable intersection theorem for closed covers of a Euclidean simplex [18]. They used the KKM lemma to provide a combinatorial proof of the Brouwer fixed point theorem (the two results being in fact equivalent). In 1961, Ky Fan extended the KKM Lemma to vector spaces of arbitrary dimensions in what be- 2 1 Introduction