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Applied Analysis Ch5 - 90 Chapter 5 Banach Spaces Many...

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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 p -norm 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 p -norm is a finite-dimensional Banach space for 1 p ≤ ∞ . 91
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92 Banach Spaces Example 5.3 The space C ([ a, b ]) of continuous, real-valued (or complex-valued) functions on [ a, b ] with the sup-norm is a Banach space. More generally, the space C ( K ) of continuous functions on a compact metric space K equipped with the sup-norm is a Banach space. Example 5.4 The space C k ([ a, b ]) of k -times continuously differentiable functions on [ a, b ] is not a Banach space with respect to the sup-norm k · k for k 1, since the uniform limit of continuously differentiable functions need not be differentiable. We define the C k -norm by k f k C k = k f k + k f 0 k + . . . + k f ( k ) k . Then C k ([ a, b ]) is a Banach space with respect to the C k -norm. Convergence with respect to the C k -norm 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 p -norm, 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 infinite-dimensional Banach space for 1 p ≤ ∞ . The sequence space p ( Z ) of bi-infinite sequences x = ( x n ) n = -∞ is defined in an analogous way. Example 5.6 Suppose that 1 p < , and [ a, b ] is an interval in R . We denote by L p ([ a, b ]) the set of Lebesgue measurable functions f : [ a, b ] R (or C ) such that Z b a | f ( x ) | p dx < , where the integral is a Lebesgue integral, and we identify functions that differ on a set of measure zero (see Chapter 12). We define the L p -norm of f by k f k p = Z b a | f ( x ) | p dx !
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