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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