Homework 13, Real Analysis

Due November 23

Problem 1

Let X be a metric space, \mathcal C_X the set of its nonempty compact subsets and d_H the Hausdorff metric.

    1. If X is discrete, then (\mathcal C_X,d_H) is discrete.
    2. Let A\subset X be a finite set of isolated points of X. Then A is an isolated point in \mathcal C_X.

Problem 2

Let X=C([-1,1]) with the inner product

\displaystyle (f,g) = \int_{-1}^1 f\bar g.

Apply the Gram-Schmidt process to the sequence f_n(x) = x^n to obtain the orthonormal polynomials p_0, p_1, p_2, p_3, such that each p_n is of degree n.

These are the first Legendre polynomials.

Problem 3

If ||\cdot||_1, ||\cdot||_2 are norms induced by inner products in \R^l, then they are equivalent: there exist constants c,C>0 such that

c||x||_1 \le ||x||_2 \le C||x||_1

for all x\in\R^l. (Hint: use the Gram-Schmidt process to construct orthonrmal bases for each inner product.)

Problem 4

Let X be an inner product space and \bar X its completion. Then \bar X is a Hilbert space.

 

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