Homework 11, Real Analysis

Due date: October 20

Problem 1

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 $A_n\subset X$ be nonempty compact sets such that $A_{n+1}\subset A_n$. Then

$\displaystyle A_n \to \bigcap_{k\ge 1} A_k$

in $(\mathcal C_X, d_H)$.

Problem 3

Two norms $||\cdot||_1, ||\cdot||_2$ are equivalent if there exist constants $c_1, c_2>0$ such that

$c_1 ||x||_1 \le ||x||_2 \le c_2 ||x||_1$

for all $x\in X$.

1. If $||\cdot||_1, ||\cdot||_2$ are equivalent, for all $\e>0$ there exist $\delta_1, \delta_2>0$ such that $B_{\delta_1}^1(x) \subset B_\e^2(x)$ and $B_{\delta_2}^2(x) \subset B_\e^1(x)$ for all $x\in X$, where $B_r^i(x)$ is the ball with respect to the norm $||\cdot||_i$.
2. If there exist $\delta,\e>0$ such that $B_\delta^1(0)\subset B_1^2(0)$ and $B_\e^2(0)\subset B_1^1(0)$, then $||\cdot||_1, ||\cdot||_2$ are equivalent.
3. If $||\cdot||_1, ||\cdot||_2$ are equivalent, then $(X,||\cdot||_1)$ and $(X,||\cdot||_2)$ are homeomorphic.
4. If $||\cdot||_1, ||\cdot||_2$ are equivalent, then $(X,||\cdot||_1)$ is complete if and only if $(X,||\cdot||_2)$ is complete.

Problem 4

1. The norms $||\cdot||_E, ||\cdot||_M, ||\cdot||_T$ in $\R^n$ are equivalent.
2. Norms induced by inner products in $\R^n$ are equivalent.
3. The $L^1, L^2$ and uniform norms in $C([0,1])$ satisfy $||f||_1 \le ||f||_2 \le ||f||_u$ for all $f\in C([0,1])$, but they are not equivalent between each other.

Problem 5

rearrangement of the series $\sum x_n$ in a norm space is a series $\sum x_{\phi(x)}$ where $\phi:\N\to\N$ is a bijection.

1. If $\sum x_n$ is absolutely convergent in a Banach space, and $\sum x_n = x$, then every rearrangement of $\sum x_n$ converges to x.
2. If $\sum x_n$ is conditionally convergent in $\R$, then, for all $x\in\R$, there exists a rearrangement of $\sum x_n$ that converges to x.
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