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Unformatted text preview: BANACH AND HILBERT SPACE REVIEW CHRISTOPHER HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but rather review the basic definitions and theorems, mostly without proof. 1. Banach Spaces Definition 1.1 (Norms and Normed Spaces) . Let X be a vector space (= linear space) over the field C of complex scalars. Then X is a normed linear space if for every f X there is a real number bardbl f bardbl , called the norm of f , such that: (a) bardbl f bardbl 0, (b) bardbl f bardbl = 0 if and only if f = 0, (c) bardbl cf bardbl =  c bardbl f bardbl for every scalar c , and (d) bardbl f + g bardbl bardbl f bardbl + bardbl g bardbl . Definition 1.2 (Convergent and Cauchy sequences) . Let X be a normed space, and let { f n } n N be a sequence of elements of X . (a) { f n } n N converges to f X if lim n bardbl f f n bardbl = 0, i.e., if > , N > , n N, bardbl f f n bardbl < . In this case, we write lim n f n = f or f n f . (b) { f n } n N is Cauchy if > , N > , m,n N, bardbl f m f n bardbl < . Definition 1.3 (Banach Spaces) . It is easy to show that any convergent sequence in a normed linear space is a Cauchy sequence. However, it may or may not be true in an arbitrary normed linear space that all Cauchy sequences are convergent. A normed linear space X which does have the property that all Cauchy sequences are convergent is said to be complete . A complete normed linear space is called a Banach space . Date : Revised September 7, 2006. c circlecopyrt 2006 by Christopher Heil. 1 2 CHRISTOPHER HEIL Example 1.4. The following are all Banach spaces under the given norms. Here p can be in the range 1 p < . L p ( R ) = braceleftBig f : R C : integraldisplay R  f ( x )  p dx < bracerightBig , bardbl f bardbl p = parenleftbiggintegraldisplay R  f ( x )  p dx parenrightbigg 1 /p . L ( R ) = braceleftbig f : R C : f is essentially bounded bracerightbig , bardbl f bardbl = esssup x R  f ( x )  . C b ( R ) = braceleftbig f L ( R ) : f is bounded and continuous bracerightbig , bardbl f bardbl = sup x R  f ( x )  , C ( R ) = braceleftbig f C b ( R ) : lim  x  f ( x ) = 0 bracerightbig , bardbl f bardbl = sup x R  f ( x )  . Remark 1.5. (a) To be more precise, the elements of L p ( R ) are equivalence classes of functions that are equal almost everywhere, i.e., if f = g a.e. then we identify f and g as elements of L p ( R ). (b) If f C b ( R ) then esssup x R  f ( x )  = sup x R  f ( x )  . Thus we regard C b ( R ) as being a subspace of L ( R ); more precisely, each element f C b ( R ) determines an equivalence class of functions that are equal to it a.e., and it is this equivalence class that belongs to L ( R )....
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 Fall '09
 HEIL

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