By Nelson Dunford, Jacob T. Schwartz

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55. Consider the space RN . Elements of RN are denoted by x = (x1 , . . , xN ), y = (y1 , . . , yN ), and so on. The norm on RN is defined by x − y = max1≤n≤N |xn − yn |. Let T : RN → RN defined by Tx = y where yk = N n=0 akn xn + bk , k = 1, . . , N. Under what conditions is T a contraction mapping ? 1 Introduction The main purpose of this book is to present basic methods and applications of Hilbert spaces. One of the most important examples of Hilbert spaces, from the point of view of both theory and applications, is the space of Lebesgue square integrable functions on RN .

Continuing this process, we construct an infinite matrix (xri rj ) such that xri rj < ε/2j+1 for all i such that i = j. In view of (b), (rj ) has a subsequence (sj ) such that limi→∞ Consider the matrix (xsi sj ). For every i ∈ N, we have ∞ j=1 xsi sj = 0. 5 Linear Mappings ∞ xsi sj = xsi si + j=1 xsi sj i=j ≥ xsi si − xsi sj i=j ≥ xsi si − xsi sj i=j ∞ j=1 xsi sj This, however, is impossible since limi→∞ proves the theorem. ε ≥ . 2 = 0. 13. (Banach–Steinhaus theorem) Let T be a family of bounded linear mappings from a Banach space X into a normed space Y .

Prove that lp is a proper vector subspace of lq whenever 1 ≤ p < q. 10. Show that any vector of R3 is a linear combination of vectors (1, 0, 0), (1, 1, 0), and (1, 1, 1). 11. Prove that every four vectors in R3 are linearly dependent. 12. Show that the functions fn (x) = xn , n = 0, 1, 2, . . , are linearly independent. 13. Show that the functions fn (x) = enx , n = 0, 1, 2, . . , are linearly independent. 14. Prove that spaces C ( ), C k (RN ), C ∞ (RN ) are infinite dimensional. 15. Denote by l0 the space of all infinite sequences of complex numbers (zn ) such that zn = 0 for all but a finite number of indices n.