banach - BANACH AND HILBERT SPACE REVIEW CHRISTOPHER HEIL...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 )....
View Full Document

Page1 / 13

banach - BANACH AND HILBERT SPACE REVIEW CHRISTOPHER HEIL...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online