201A_F10_hw3_sol

201A_F10_hw3_sol - 201A, Fall 10, Thomases Homework 3...

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201A, Fall ’10, Thomases Homework 3 Mihaela Ifrim 1. Let X be a normed linear space. A series in x n in X is absolutely convergent if b x n b converges to a Fnite value in R . Prove X is a Banach space if and only if every absolutely convergent series converges. Proof. Let X be a Banach space and suppose that ( x n ) is absolutely convergent; i. e., b x n b = y for some Fnite y . ±or any ε > 0, let N be su²ciently large so that n = N b x n b < ε . If m,n N , then b i = m x i i = m ( x i ) b = b m i = n ( x i ) b < m i = n b x i b < ε (applying the triangle inequality). Therefore, n x n is Cauchy and hence convergent. We conclude that every absolutely convergent series in X converges. Conversely, let X be a normed linear space in which every absolutely convergent series converges, and suppose that { x n } n is a Cauchy se- quence. ±or each k N , choose n k such that b ( x m x n ) b < 2 - k for m,n n k . In particular, b x n k +1 x n k b < 2 - k . If we deFne y 1 = x n 1 and y k +1 = x n k +1 x n k for k 1, it follows that ( b y n b ) ≤ b x n 1 b + 1 : i. e., ( y n ) is absolutely convergent, and hence convergent. More explicitly n b y n b ) = k b x n k +1 x n k b + b x n 1 b ≤ b x n 1 b + k 1 2 k = b x n 1 b + 1. However, this implies that b y n b ≤ ∞ . But we considered that every absolute convergent series is convergent, which means that y n ≤ ∞ . But the sequence of partial sums of y n is a sub- sequence of x n . Since x n is Cauchy and has a convergent subsequence, then it will aslo converge to the same limit as the subsequence. Hence we have started with a random Cauchy sequence in X and we got that
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This note was uploaded on 01/29/2012 for the course MATH 34 taught by Professor Wiley during the Spring '11 term at UC Merced.

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201A_F10_hw3_sol - 201A, Fall 10, Thomases Homework 3...

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