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Homework 12, Real Analysis

Due date: October 27

Problem 1

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

  1. $latex x_n + y_n \to x + y$;

  2. $latex \lambda_n x_n \to \mu x$ for all sequences $latex \lambda_n\to\lambda$ in $latex \K$.

Problem 2

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

  1. $latex \lim ||x_n-x|| = 0$ implies $latex \lim||x_n||=||x||$.

  2. For $latex x,y\in X,\lambda\in\R$, find $latex \lim \big( ||(n+\lambda)x + y|| - ||nx+y||\big)$.

Problem 3

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

Problem 4

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

  1. $latex x_n$ converges.

  2. Give an example where the previous statement is false if X is not complete.

Problem 5

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

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

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