Ir al contenido principal

Due August 21

Problem 1

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

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

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

Problem 2

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

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

Problem 3

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

Problem 4

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

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

Problem 5

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

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