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By Alexander J. W.

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Therefore x is an interior point of S. We conclude that S is open. I The proofs of the following two complementary results are left to the reader. Proposition 2. The intersection of a finite number of open sets is open; the union of an arbitrary collection of open sets is open. Proposition 3. The union of a finite number of closed sets is closed; the intersection of an arbitrary collection of closed sets is closed. We now have two topological operations, taking closures and taking interiors, that can bl~ applied to sets in normed space.

Each x. has a unique representation as For arbitrary n, m Ilx. - xmll = II itp7 - Aj)el\1 ;;:: IA~ - A~lo for each k, I ~ k ~ N. Since Ilx. - xmll-+ 0, each IA7 - A~I-+ O. Thus, {A~},~)= 1 is a Cauchy sequence of scalars and hence convergent to a scalar Ak' Let x = Lk=1 Akek' Obviously, x EM. We show that x. -+ x. For all n, we have Ilx. - xii = II k=1 f(Ak - Ak)ekll s:; N· max IAk - Akl' Ilekll, 1skSN but since IAk - Akl-+ 0 for all k, XEM. I Ilx. - xii -+ O. 13 Extreme Values of Functionals and Compactness Optimization theory is largely concerned with the maximization or minimization of real functionals over a given subset; indeed, a major portion of this book is concerned with principles for finding the points at which a given functional attains its maximum.

A finite set S of linearly independent vectors is said to be a basis for the space X if S generates X. A vector space having a finite basis is said to be finite dimensional. All other vector spaces are said to be infinite dimensional. rize a finite-dimensional space by the number of elements in a basis. Thus, a space with a basis consisting of n elements is referred to as n-diml~nsional space. This practice would be undesirable on grounds of ambiguity if the number of elements in a basis for a space were not unique.

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