Homework 12, Real Analysis

Posted by Ricardo A. Sáenz
on Oct, 20, 2017
in Real Analysis, 2017
Blog No Comments.

Due date: October 27

Problem 1

Let $x_n\to x, y_n\to y$ be convergent sequences in the normed space $(X,||\cdot||)$ over $\K$ . Then

$x_n + y_n \to x + y$ ;
$\lambda_n x_n \to \mu x$ for all sequences $\lambda_n\to\lambda$ in $\K$ .

Problem 2

Let $(X,||\cdot||)$ be a normed space over $\K$ .

$\lim ||x_n-x|| = 0$ implies $\lim||x_n||=||x||$ .
For $x,y\in X,\lambda\in\R$ , find $\lim \big( ||(n+\lambda)x + y|| - ||nx+y||\big)$ .

Problem 3

Let X be a Banach space, $x_n\in X$ , $||x_n||=1$ for all n, and $\lambda_n\in\K$ . Discuss the validity of the statement: $\sum \lambda_n x_n$ converges if and only if $\sum |\lambda_n|<\infty$ .

Problem 4

Let $x_n$ be a sequence in a Banach space X such that, for all $\e>0$ , there exists a convergent sequence $y_n$ such that $||x_n - y_n||<\e$ for all n.

$x_n$ converges.
Give an example where the previous statement is false if X is not complete.

Problem 5

The normed space $(X,||\cdot||)$ is a Banach space if and only if the unit sphere

$\mathbb S = \{ x\in X: ||x||=1\}$

equipped with the metric $d(x,y)=||x-y||$ is complete.

Related

Post Tagged with Tareas

Comments & Responses

Leave a Reply Cancel reply

%d bloggers like this: