Due September 18th
Problem 1
State whether the following families of functions are equicontinuous, pointwise bounded, or both.
- $latex \{\sin nx\}_{n\ge1}$ in $latex C([0,2\pi])$
- $latex \{x^n\}_{n\ge1}$ in $latex C([0,1])$
- $latex \{\dfrac{x^n}{n}\}_{n\ge1}$ in $latex C([0,2])$
Problem 2
Let $latex f_n:[a,b]\to\R$ be a monotone sequence of continuous functions, pointwise convergent to the continuous function $latex f:[a,b]\to\R$. Then $latex f_n\rightrightarrows f$ on $latex [a,b]$.
Problem 3
Let $latex \Omega\subset\R^m$ be open and $latex (f_n)$ an equicontinuous sequence of functions that converges pointwise in $latex \Omega$. Then $latex (f_n)$ converges uniformly on compact subsets of $latex \Omega$.
Problem 4
Let $latex K:[0,1]\times[0,1]\to\R$ be continuous and define the operator $latex L:C([0,1])\to C([0,1])$ by
$latex \displaystyle Lf(x) = \int_0^1 K(x,y)f(y) dy.$
Then the closure $latex \overline{L(B_1(0))}$ of the image of the unit ball under $latex L$ is compact in $latex C([0,1])$.
$latex L$ is an example of a compact operator.
Problem 5
Let $latex (X,d)$ be a compact metric space and
$latex \mathcal L = \{ f\in C(X,\R^l): ||f||_u\le 1, |f(x) - f(y)|\le d(x,y)\quad \forall x,y\in X\}.$
Then $latex \mathcal L$ is compact.
Comentarios
Publicar un comentario