Homework 8, Complex Analysis

Problem 1

Consider the function on $latex [0,\infty)$ given by $latex \psi(t) = \dfrac{t}{1+t}$.

1. $latex \psi$ is concave and increasing
2. $latex \psi$ is continuous at $latex 0$
3. For any $latex \varepsilon > 0$ there exists $latex \delta > 0$ such that $latex \psi(t) < \delta$ implies $latex 0\le t < \varepsilon$

Problem 2

Let $latex d_j$ be a sequence of quasimetrics on $latex X$ that separates points, and $latex d$ the metric constructed from the $latex d_j$ as in class. Let, for each positive integer $latex N$, $latex x\in X$ and $latex \varepsilon>0$,

$latex B_N(x,\varepsilon) = \{ y\in X: d_j(x,y) < \varepsilon, j = 1, 2, \ldots, N\}.$

Then $latex B_N(x,\varepsilon)$ is open with respect to the metric $latex d$.

Problem 3

$latex S\in X$ is totally bounded in $latex (X,d)$ if and only if for any $latex \varepsilon > 0$ and any positive integer $latex N$ there exist $latex x_1, x_2, \ldots, x_m$ such that

$latex \displaystyle S \subset \bigcup_{l=1}^m B_N(x_l,\varepsilon)$.

Problem 4

Let $latex V, U_j\subset \mathbb C$ open such that $latex V = \bigcup_{j=1}^\infty U_j$. Suppose $latex F:V\to\mathbb C$ is such that $latex F|_{U_j}$ is continuous in $latex U_j$. Then $latex F$ is continuous in $latex V$.

Problem 5

Let $latex V\subset\mathbb C$ open and $latex S\subset C(V)$ pointwise bounded and pointwise equicontinuous. Then $latex S$ is uniformly bounded on any compact $latex K\subset V$.