### Archive

Posts Tagged ‘compact’

## Dini’s theorem and related problems

Let ${ a be two real numbers and ${ f_n :[a,b ]\rightarrow \Bbb{R}}$ a sequence of continuous functions which converge pointwise to a continuous function ${ f}$.

1. Dini’s Theorem. Suppose that the sequence ${ (f_n)}$ has the property that ${ f_n \leq f_{n+1}}$. Prove that the convergence ${ f_n \rightarrow f}$ is uniform.

2. Suppose that every function ${ f_n}$ is increasing. Prove that the convergence ${ f_n \rightarrow f}$ is uniform.

3. If every function ${ f_n}$ is convex on ${ [a,b]}$, prove that the convergence ${ f_n \rightarrow f}$ is uniform on every compact interval in ${ (a,b)}$.

Categories: Analysis, Real Analysis

## Agreg 2012 Analysis Part 4

A Fixed Point Theorem

This parts wishes to extend the next result (which can be used without proof) to infinite dimension.

Theorem. (Browuer) Consider ${F}$ a finite dimensional normed vector space. Consider ${C\subset F}$ a convex, closed, bounded non-void set. If ${f:C\rightarrow C}$ is a continuous application, then ${f}$ has a fixed point in ${C}$.

1. In ${\ell^2(\Bbb{N})}$ endowed with ${\|\cdot \|_2}$ we consider the following application:

$\displaystyle f: B(0,1) \subset \ell^2(\Bbb{N}) \rightarrow \ell^2(\Bbb{N})$

$\displaystyle f(x) =(\sqrt{1-\|x\|_2},x_0,x_1,...).$

Prove that ${f}$ is continuous with values in the unit sphere of ${\ell^2(\Bbb{N})}$, but ${f}$ does not admit any fixed points.

2. Consider ${E}$ a normed vector space, ${B}$ a closed, bounded non-void subset of ${E}$ and ${f:B \rightarrow E}$ a compact application (not necessarily linear). (a compact application maps bounded sets into relatively compact sets)

i) Let ${n \in \Bbb{N}\setminus \{0\}}$. We can cover ${\overline{f(B)}}$ (which is compact) by a finite number ${N_n}$ of open balls of radius ${\frac{1}{n}}$: ${\overline{f(B)} \subset \displaystyle \bigcup_{i=1}^{N_n} \mathring{B}(y_i,\frac{1}{n})}$ with ${y_i \in \overline{f(B)}}$ for every ${i}$. For ${y \in E}$ we define

$\displaystyle \psi(y) =\begin{cases} \frac{1}{n}-\|y-y_i\| & \text{ if } y \in B(y_i,\frac{1}{n}) \\ 0 & \text{ otherwise} \end{cases}$

Prove that ${\Psi : y \in \overline{f(B)} \mapsto \sum_{i=1}^{N_n} \psi_i(y)}$ is continuous and that there exists ${\delta>0}$ such that for ${y \in \overline{f(B)}}$ we have ${\Psi(y)\geq \delta}$.

ii) We introduce the application ${f_n : B \rightarrow E}$ defined by

$\displaystyle f_n(x)= \left( \sum_{i=1}^{N_n} \psi_i(f(x))\right)^{-1} \sum_{i=1}^{N_n} \psi_i(f(x))y_i.$

Prove that for every ${x \in B}$ we have ${\|f(x)-f_n(x) \|\leq \frac{1}{n}}$.

Categories: Analysis, Fixed Point

## Compact operator maps weakly convergent sequences into strong convergent sequences

Suppose $T \in \mathcal{L}(E,F)$ is a compact operator and $(u_n)$ is a sequence in $E$ such that $u \rightharpoonup u$ (i.e. converges weakly). Prove that $Tu_n \to Tu$ strongly in $F$.
Suppose $K \subset \mathbb{R}^n$ is a compact convex set and $f: K \to K$ is a function satisfying $\| f(x)-f(y) \| \leq \|x -y\|,\ \forall x,y \in K$. Prove that $f$ has a fixed point ( i.e. $\exists x \in K$ with $f(x)=x$). Read more…