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Homework 13, Real Analysis

Due November 23

Problem 1

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

    1. If X is discrete, then $latex (\mathcal C_X,d_H)$ is discrete.

    2. Let $latex A\subset X$ be a finite set of isolated points of X. Then A is an isolated point in $latex \mathcal C_X$.

Problem 2

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

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

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

These are the first Legendre polynomials.

Problem 3

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

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

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

Problem 4

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