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LpCompleteness

# LpCompleteness - p IS COMPLETE Let 1 p and recall the...

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p IS COMPLETE Let 1 p ≤ ∞ , and recall the definition of the metric space p : For 1 p < , p = ( sequences a = ( a n ) n =1 in R such that X n =1 | a n | p < ) ; whereas consists of all those sequences a = ( a n ) n =1 such that sup n N | a n | < . We defined the p -norm as the function k · k p : p [0 , ) , given by k a k p = X n =1 | a n | p ! 1 /p , for 1 p < , and k a k = sup n N | a n | . In class, we showed that the function d p : p × p [0 , ) given by d p ( a , b ) = k a - b k p is actually a metric. We now proceed to show that ( p , d p ) is a complete metric space for 1 p ≤ ∞ . For convenience, we will work with the case p < , as the case p = requires slightly different language (although the same ideas apply). Suppose that a 1 , a 2 , a 3 , . . . is a Cauchy sequence in p . Note, each term a k in the se- quence is a point in p , and so is itself a sequence: a k = ( a k 1 , a k 2 , a k 3 , . . . ) . Now, to say that ( a k ) k =1 is a Cauchy sequence in p is precisely to say that > 0 K N s.t. k, m K, k a k - a m k p < . That is, for given > 0 and sufficiently large k, m , we have X n =1 | a k n - a m n | p = k a k - a m k p p < p . Now, the above series has all non-negative terms, and hence is an upper bound for any fixed term in the series. That is to say, for fixed n 0 N , | a k n 0 - a m n 0 | ≤ X n =1 | a k n - a m n | p < p , and so we see that the sequence ( a k n 0 ) k =1 is a Cauchy sequence in R . But we know that R is a complete metric space, and thus there is a limit a n 0 R to this sequence. This holds for each n 0 N . The following diagram illustrates what’s going on. a 1 = a 1 1 a 1 2 a 1 3 a 1 4 · · · a 2 = a 2 1 a 2 2 a 2 3 a 2 4 · · · a 3 = a 3 1 a 3 2 a 3 3 a 3 4 · · · a 4 = a 4 1 a 4 2 a 4 3 a 4 4 · · · . . . . . . . . . . . . . . . a 1 a 2 a 3 a 4 · · · So, we have shown that, in this p -Cauchy sequence of horizontal sequences, each vertical sequence actually converges. Hence, there is a sequence a = ( a 1 , a 2 , a 3 , a 4 , . . . ) to which a k converges” in a vague sense. The sense is the “point-wise convergence” along vertical 1

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2 lines in the above diagram.
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LpCompleteness - p IS COMPLETE Let 1 p and recall the...

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