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.

Post Tagged with