Author Archives: Ricardo A. Sáenz

Homework 16, Real Analysis

Due date: November 24 Problem 1 If the measurable with , then Explain the condition . Problem 2 Let and . Then Problem 3 There exists a positive continuous such that If is uniformly continuous, then Problem 4 If and . Then F is uniformly continuous.

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Homework 15, Real Analysis

Due date: November 17 Problem 1 For each , let . Then Problem 2 For , the sequence as . Problem 3 Let with . The Fourier series of f converges uniformly to f. Problem 4 Let and the open set If E is compact, . However, the previous conclusion may be false if either E is closed and […]

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Homework 14, Real Analysis

Due date: November 10 Problem 1 Let Y be a vector subspace of the normed space X. Then its closure is also a vector subspace of X. Problem 2 Let with the inner product Apply the Gram-Schmidt process to the sequence to obtain the orthonormal polynomials , such that each is of degree n. These are the first Legendre polynomials. […]

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Homework 13, Real Analysis

Due date: November 3 Problem 1 Let be a sequence of quadratic polynomials such that Then the coefficient sequences all converge to zero. Problem 2 For , let be the space of polynomials of degree at most r. If converge uniformly to f in [0,1], then . The polynomials  converge uniformly on [0,1], but their limit is not […]

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Homework 12, Real Analysis

Due date: October 27 Problem 1 Let be convergent sequences in the normed space over . Then ; for all sequences in . Problem 2 Let  be a normed space over . implies . For , find . Problem 3 Let X be a Banach space, , for all n, and . Discuss the validity of the […]

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