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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