Tarea 2, Análisis real

Due August 21

Problem 1

Let $latex (X,||\cdot||)$ be a normed vector space and $latex x_n, y_n$ sequences in $latex X$ such that $latex x_n\to x$ and $latex y_n \to y$. Then

$latex \lambda x_n + \mu y_n \to \lambda x + \mu y$

for any $latex \lambda,\mu\in\mathbb K$.

Problem 2

1. If $latex (X,d)$ and $latex (X,d')$ be homeomorphic metric spaces, then they have the same convergence sequences.


2. However, there exists homeomorphic metric spaces $latex (X,d), (X,d')$ such that only one of them is complete.

Problem 3

If $latex (X,||\cdot||)$ and $latex (X,||\cdot||')$ are homeomorphic, then $latex (X,||\cdot||)$ is complete if and only if $latex (X,||\cdot||')$ is complete.

Problem 4

1. Let $latex  f_n$ be the sequence of functions in $latex C([0,1])$ given by
   $latex \displaystyle f_n(x) = \begin{cases}\sqrt n & 0\le x<\dfrac{1}{n}\\\dfrac{1}{\sqrt x} & \dfrac{1}{n}\le x\le 1.\end{cases}$
   Then $latex f_n$ is a Cauchy sequence in $latex (C([0,1]),||\cdot||_1)$ that does not converge. Thus, $latex (C([0,1]),||\cdot||_1)$ is not complete.


2. Is $latex (C([0,1]),||\cdot||_2)$ complete?

Problem 5

A reordering of the series $latex \sum x_n$ in a Banach space $latex X$ is a series $latex \sum x_{\phi(n)}$ where $latex \phi:\Z_+\to\Z_+$ is a bijection.

If $latex \sum x_n$ converges absolutely, and converges to $latex x$, then any reordering of $latex \sum x_n$ converges to $latex x$.