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Posts Tagged ‘compact’

Dini’s theorem and related problems

May 13, 2014 Leave a comment

Let { a<b} 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)}.

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Agreg 2012 Analysis Part 4

October 20, 2013 Leave a comment

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}}.

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Compact operator maps weakly convergent sequences into strong convergent sequences

October 6, 2011 Leave a comment

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.

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Fixed point theorem

January 12, 2010 Leave a comment

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…

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