Excursion to Functional Analysis

Abstract

In modern analysis, most of the problems in science and engineering have solutions in infinite-dimensional spaces of functions, where closed bounded sets are usually not compact, the measure theory is of limited use, linear operators may not have eigenvalues even in the complex case, or even reasonable continuous functions do not attain their extrema on closed bounded sets, just to mention some difficulties. Thus we encounter there problems additional to those we handled in the first chapters of this text. Functional analysis helps in solving these problems by providing additional powerful tools that work in general contexts, allowing in this way to treat a large variety of situations looking at them from a unifying point of view.