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Homework 4, Real Analysis

Due date: September 1

Problem 1

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

Problem 2

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

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

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

Such operator is called a compact operator.

Problem 3

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

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

is compact.

Problem 4

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

Problem 5

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

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

is compact.