linear analysis2019Lect1-3.pdf - NOTE ON MMAT 5010 LINEAR ANALYSIS(2019 1ST TERM CHI-WAI LEUNG 1 Lecture 1 Throughout this note we always denote K by

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NOTE ON MMAT 5010: LINEAR ANALYSIS (2019 1ST TERM) CHI-WAI LEUNG 1. Lecture 1 Throughout this note, we always denote K by the real field R or the complex field C . Let N be the set of all natural numbers. Also, we write a sequence of numbers as a function x : { 1 , 2 , ... } → K or x i := x ( i ) for i = 1 , 2 ... . Definition 1.1. Let X be a vector space over the field K . A function k · k : X R is called a norm on X if it satisfies the following conditions. (i) k x k ≥ 0 for all x X and k x k = 0 if and only if x = 0 . (ii) k αx k = | α |k x k for all α K and x X . (iii) k x + y k ≤ k x k + k y k for all x, y X . In this case, the pair ( X, k · k ) is called a normed space. Remark 1.2. Recall that a metric space is a non-empty set Z together with a function, ( called a metric ), d : Z × Z R that satisfies the following conditions: (i) d ( x, y ) 0 for all x, y Z ; and d ( x, y ) = 0 if and only if x = y . (ii) d ( x, y ) = d ( y, x ) for all x, y Z . (iii) d ( x, y ) d ( x, z ) + d ( z, y ) for all x, y and z in Z . For a normed space ( X, k·k ) , if we define d ( x, y ) := k x - y k for x, y X , then X becomes a metric space under the metric d . The following examples are important classes in the study of functional analysis. Example 1.3. Consider X = K n . Put k x k p := ( n X i =1 | x i | p ) 1 /p and k x k := max i =1 ,...,n | x i | for 1 p < and x = ( x 1 , ..., x n ) K n . Then k · k p (called the usual norm as p =2) and k · k (called the sup -norm) all are norms on K n . Example 1.4. Put c 0 := { ( x ( i )) : x ( i ) K , lim | x ( i ) | = 0 } (called the null sequnce space) and := { ( x ( i )) : x ( i ) K , sup i | x ( i ) | < ∞} . Then c 0 is a subspace of . The sup -norm k · k on is defined by k x k := sup i | x ( i ) | Date : September 20, 2019. 1
2 CHI-WAI LEUNG for x . Let c 00 := { ( x ( i )) : there are only finitly many x ( i ) ’s are non-zero } . Also, c 00 is endowed with the sup -norm defined above and is called the finite sequence space. Example 1.5. For 1 p < , put p := { ( x ( i )) : x ( i ) K , X i =1 | x ( i ) | p < ∞} . Also, p is equipped with the norm k x k p := ( X i =1 | x ( i ) | p ) 1 p for x p . Then k · k p is a norm on p (see [2, Section 9.1] ). Example 1.6. Let C b ( R ) be the space of all bounded continuous R -valued functions f on R . Now C b ( R ) is endowed with the sup -norm, that is, k f k = sup x R | f ( x ) | for every f C b ( R ) . Then k · k is a norm on C b ( R ) . Also, we consider the following subspaces of C b ( X ) . Let C 0 ( R ) ( resp. C c ( R ) ) be the space of all continuous R -valued functions f on R which vanish at infinity (resp. have compact supports), that is, for every ε > 0 , there is a K > 0 such that | f ( x ) | < ε (resp. f ( x ) 0 ) for all | x | > K . It is clear that we have C c ( R ) C 0 ( R ) C b ( R ) . Now C 0 ( R ) and C c ( R ) are endowed with the sup -norm k · k . From now on, we always let X be a normed sapce. Definition 1.7. We say that a sequence ( x n ) in X converges to an element a X if lim k x n - a k = 0 , that is, for any ε > 0 , there is N N such that k x n - a k < ε for all n N .

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