# Homework 12, Real Analysis

## 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

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

### Problem 2

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

1. $\lim ||x_n-x|| = 0$ implies $\lim||x_n||=||x||$.
2. 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.

1. $x_n$ converges.
2. 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.

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