Titlebook: Real and Functional Analysis; Serge Lang Textbook 1993Latest edition Springer-Verlag New York, Inc. 1993 Banach Space.Distribution.Hilbert

Discrete

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Continuous Functions on Compact Setsand also the notion of convergent sequence (having a limit). If every Cauchy sequence converges, then . is said to be ., and is also called a .. A closed subspace of a Banach space is complete, hence it is also a Banach space.

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Banach Spacesctions, and the most frequent test for convergence (in fact absolute convergence) is the standard one:Let {..} be a sequence of numbers ≧ 0 such that ∑ .. converges. If |..| .. for all ., then ∑ .. converges.

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978-1-4612-6938-0Springer-Verlag New York, Inc. 1993

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SetsWe assume that the reader understands the meaning of the word “set”, and in this chapter, summarize briefly the basic properties of sets and operations between sets. We denote the empty set by ?. A subset .′ of . is said to be . if .′ ≠ .. We write .′ ?. or . ? .′ to denote the fact that .′ is a subset of ..

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Duality and Representation TheoremsConsider first complex valued functions. We let ?.(.) be the set of all functions . on . that are limits almost everywhere of a sequence of step functions (i.e. .-measurable), and such that |.|. lies in ?.. Thus

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