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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 ∀ ε > , ∃ N > , ∀ n ≥ N, bardbl f − f n bardbl < ε. In this case, we write lim n →∞ f n = f or f n → f . (b) { f n } n ∈ N is Cauchy if ∀ ε > , ∃ N > , ∀ m,n ≥ N, bardbl f m − f n bardbl < ε. 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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This note was uploaded on 08/25/2011 for the course MATH 7338 taught by Professor Heil during the Fall '09 term at Georgia Tech.
 Fall '09
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