The present book is the outcome of efforts to introduce topological connectedness as one of the basic tools for the study of necessary conditions for an extremum. Apparently this mograph is the first book in the theory of maxima and minima where topological connectedness is used so widely for this purpose. Its application permits us to obtain new results in this sphere and to consider the classical results from a nstandard point of view. Regarding the style of the present book it should be remarked that it is comparatively elementary. The author has made constant efforts to make the book as self-contained as possible. Certainly, familiarity with the basic facts of topology, functional analysis, and the theory of optimization is assumed. The book is written for applied mathematicians and graduate students interested in the theory of optimization and its applications. We present the synthesis of the well kwn Dybovitskii'-Milyutin ap- proach for the study of necessary conditions for an extremum, based on functional analysis, and topological methods. This synthesis allows us to show that in some cases we have the following important result: if the Euler equation has n trivial solution at a point of an extremum, then some inclusion is valid for the functionals belonging to the dual space. This general result is obtained for an optimization problem considered in a lin- ear topological space. We also show an application of our result to some problems of nlinear programming and optimal control.