Tarea 6, Análisis real

Due September 18th

Problem 1

State whether the following families of functions are equicontinuous, pointwise bounded, or both.

1. $latex \{\sin nx\}_{n\ge1}$ in $latex C([0,2\pi])$


2. $latex \{x^n\}_{n\ge1}$ in $latex C([0,1])$


3. $latex \{\dfrac{x^n}{n}\}_{n\ge1}$ in $latex C([0,2])$

Problem 2

Let $latex f_n:[a,b]\to\R$ be a monotone sequence of continuous functions, pointwise convergent to the continuous function $latex f:[a,b]\to\R$. Then $latex f_n\rightrightarrows f$ on $latex [a,b]$.

Problem 3

Let $latex \Omega\subset\R^m$ be open and $latex (f_n)$ an equicontinuous sequence of functions that converges pointwise in $latex \Omega$. Then $latex (f_n)$ converges uniformly on compact subsets of $latex \Omega$.

Problem 4

Let $latex K:[0,1]\times[0,1]\to\R$ be continuous and define the operator $latex L:C([0,1])\to C([0,1])$ by

$latex \displaystyle Lf(x) = \int_0^1 K(x,y)f(y) dy.$

Then the closure $latex \overline{L(B_1(0))}$ of the image of the unit ball under $latex L$ is compact in $latex C([0,1])$.

$latex L$ is an example of a compact operator.

Problem 5

Let $latex (X,d)$ be a compact metric space and

$latex \mathcal L = \{ f\in C(X,\R^l): ||f||_u\le 1, |f(x) - f(y)|\le d(x,y)\quad \forall x,y\in X\}.$

Then $latex \mathcal L$ is compact.