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Homework 11, Real Analysis

Due date: October 20

Problem 1

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

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

in $latex (\mathcal C_X, d_H)$.

Problem 3

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

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

for all $latex x\in X$.

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

  2. If there exist $latex \delta,\e>0$ such that $latex B_\delta^1(0)\subset B_1^2(0)$ and $latex B_\e^2(0)\subset B_1^1(0)$, then $latex ||\cdot||_1, ||\cdot||_2$ are equivalent.

  3. If $latex ||\cdot||_1, ||\cdot||_2$ are equivalent, then $latex (X,||\cdot||_1)$ and $latex (X,||\cdot||_2)$ are homeomorphic.

  4. If $latex ||\cdot||_1, ||\cdot||_2$ are equivalent, then $latex (X,||\cdot||_1)$ is complete if and only if $latex (X,||\cdot||_2)$ is complete.

Problem 4

  1. The norms $latex ||\cdot||_E, ||\cdot||_M, ||\cdot||_T$ in $latex \R^n$ are equivalent.

  2. Norms induced by inner products in $latex \R^n$ are equivalent.

  3. The $latex L^1, L^2$ and uniform norms in $latex C([0,1])$ satisfy $latex ||f||_1 \le ||f||_2 \le ||f||_u$ for all $latex f\in C([0,1])$, but they are not equivalent between each other.

Problem 5

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

  1. If $latex \sum x_n$ is absolutely convergent in a Banach space, and $latex \sum x_n = x$, then every rearrangement of $latex \sum x_n$ converges to x.

  2. If $latex \sum x_n$ is conditionally convergent in $latex \R$, then, for all $latex x\in\R$, there exists a rearrangement of $latex \sum x_n$ that converges to x.