Homework 6, Real Analysis

Due September 28

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

1. 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.


2. 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 3

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

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 $latex \Omega$.