Ir al contenido principal

## Due September 18th

### Problem 1

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

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

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

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

### Problem 2

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

### Problem 3

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

### Problem 4

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

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

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

$L$ is an example of a compact operator.

### Problem 5

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

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

Then $\mathcal L$ is compact.