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## Due date: September 1

### Problem 1

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

### Problem 2

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

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

Then, the image of the closed ball $\bar B_1(0)$ in $C([0,1])$ under $\mathscr L$ is compact.

Such operator is called a compact operator.

### Problem 3

Let $w:[0,1]\to\R$ be continuous. Then the operator

$\displaystyle \mathscr Lf(x) = \int_0^x f(t) w(t)dt$

is compact.

### Problem 4

Let $\Omega\subset\R^m$ be open and $f_n:\Omega\to\R$ an equicontinuous sequence of functions that converges pointwise. Then $f_n$ converges uniformly on each compact subset of $\Omega$.

### Problem 5

Let X be a compact metric space. The family $\mathscr F\subset C(X,\R^l)$ given by

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

is compact.