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Unformatted text preview: 90 Chapter 5 Banach Spaces Many linear equations may be formulated in terms of a suitable linear operator acting on a Banach space. In this chapter, we study Banach spaces and linear oper ators acting on Banach spaces in greater detail. We give the definition of a Banach space and illustrate it with a number of examples. We show that a linear operator is continuous if and only if it is bounded, define the norm of a bounded linear op erator, and study some properties of bounded linear operators. Unbounded linear operators are also important in applications: for example, differential operators are typically unbounded. We will study them in later chapters, in the simpler context of Hilbert spaces. 5.1 Banach spaces A normed linear space is a metric space with respect to the metric d derived from its norm, where d ( x, y ) = k x y k . Definition 5.1 A Banach space is a normed linear space that is a complete metric space with respect to the metric derived from its norm. The following examples illustrate the definition. We will study many of these examples in greater detail later on, so we do not present proofs here. Example 5.2 For 1 p < , we define the pnorm on R n (or C n ) by k ( x 1 , x 2 , . . . , x n ) k p = (  x 1  p +  x 2  p + . . . +  x n  p ) 1 /p . For p = , we define the , or maximum, norm by k ( x 1 , x 2 , . . . , x n ) k = max { x 1  ,  x 2  , . . . ,  x n } . Then R n equipped with the pnorm is a finitedimensional Banach space for 1 p . 91 92 Banach Spaces Example 5.3 The space C ([ a, b ]) of continuous, realvalued (or complexvalued) functions on [ a, b ] with the supnorm is a Banach space. More generally, the space C ( K ) of continuous functions on a compact metric space K equipped with the supnorm is a Banach space. Example 5.4 The space C k ([ a, b ]) of ktimes continuously differentiable functions on [ a, b ] is not a Banach space with respect to the supnorm k k for k 1, since the uniform limit of continuously differentiable functions need not be differentiable. We define the C knorm by k f k C k = k f k + k f k + . . . + k f ( k ) k . Then C k ([ a, b ]) is a Banach space with respect to the C knorm. Convergence with respect to the C knorm is uniform convergence of functions and their first k deriva tives. Example 5.5 For 1 p < , the sequence space p ( N ) consists of all infinite sequences x = ( x n ) n =1 such that X n =1  x n  p < , with the pnorm, k x k p = X n =1  x n  p ! 1 /p . For p = , the sequence space ( N ) consists of all bounded sequences, with k x k = sup { x n   n = 1 , 2 , . . . } . Then p ( N ) is an infinitedimensional Banach space for 1 p . The sequence space p ( Z ) of biinfinite sequences x = ( x n ) n = is defined in an analogous way....
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This note was uploaded on 03/23/2012 for the course AMATH 567 taught by Professor A.g during the Fall '11 term at University of Washington.
 Fall '11
 A.G

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