Extremal Problems

Abstract

The common denominator of the three parts forming this chapter is the search for extrema of real-valued functionals φ over various subsets Ω of usually infinite-dimensional Banach spaces X under various conditions on φ, Ω and X. The need for an abstract ordering of ideas concerning the minimization of certain functions on ‘collections of functions’ (the calculus of variations) is one of the origins of what is called functional analysis today, but not the only one, as you are led to believe in some books on the variational calculus. For the history of the latter you may consult Goldstine [1] and some remarks in Young [1]. If you are interested in the history of functional analysis, you should not miss the booklet of F. Riesz [1] or the recent retrospection of Dieudonné [2], the fourth chapter of which draws from the work of Riesz — to mention just two references. These days optimization theory is no doubt the most popular mathematical discipline for the ‘users’ (besides probability theory), if the incredible flood of books on the subject is a measure of popularity (‘that’s what sells’). So it may be useful to have an abstract of the abstraction, showing in particular how some techniques from other chapters may be used here.

... we reflect that the majorities of the ideas we deal with were conceived by others, often centuries ago. In a great measure it is really the intelligence of other people that confronts us in science.

Ernst Mach

Every problem in the calculus of variations has a solution, provided the word solution is suitably understood.

David Hilbert

The real mathematician is an enthusiast per se. Without enthusiasm no mathematics.

Novalis