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Due September 28

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

1. 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.

2. 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 3

Let $F_1\subset F_2\subset \ldots$ an increasing sequence of compact subsets of a metric space, and let $K \subset\subset \bigcup F_n$. Then there exists p such that $K\subset F_p$.

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 $\Omega$.