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Mostrando las entradas de octubre, 2015

### El teorema de punto fijo de Schauder

The proof of Schauder's fixed point theorem we saw in class was, sadly, incomplete, as the approximating functions defined on the finite dimensional convex sets were not well defined. Here is a correct proof of the theorem. Theorem (Schauder).   Let $V$ be a compact convex subset of the Banach space $X$ and $f:V\to V$ continuous. Then $f$ has a fixed point. Proof. For a given $n\in\Z_+$, let $x_1, \ldots, x_k\in V$ be such that $\displaystyle V\subset\bigcup_i^k B_{1/n}(x_i)$. Such $x_i$ exist because $V$ is compact. Define, for each $i$, the functions $\lambda_i:V\to\R$ by $\lambda_i(x) = \begin{cases}1/n-||x_i - x||& x\in B_{1/n}(x_i)\\0 & \text{otherwise.}\end{cases}$ The functions $\lambda_i$ are continuous and $\sum_i \lambda_i(x)\not=0$ for every $x\in V$. Thus, if we define $\pi_n(x) = \dfrac{\sum \lambda_i(x)x_i}{\sum\lambda_i(x)}$, $\pi_n$ maps $V$ into the

### Tarea 12, Análisis real

Due October 30 Problem 1 Let $A\subset X$ be connected. Then $\bar A$ is connected. Problem 2 If $A\subset X$ is connected and $A\subset B\subset \bar A$, then $B$ is connected. Problem 3 $X$ is connected if and only if every continuous $f:X\to Y$ into a discrete space $Y$ is constant. Problem 4 If $A$ is convex, then $\bar A$ is convex. Problem 5 Prove $B_r(x_0)$ is convex from the fact that $B_1(0)$ is convex.

### Tarea 10, Análisis real

Due October 16 Problem 1 If $X$ is discrete, then $(\mathcal C_X,d_H)$ is discrete. Problem 2 Let $A\subset X$ be a finite set of isolated points in $X$. Then $A$ is isolated in $\mathcal C_X$. Problem 3 Let $A_n$ be a decreasing sequence of nonempty compact sets in $X$. Then $\displaystyle \lim A_n = \bigcap_{n\ge 1} A_n$ in $\mathcal C_H$. Problem 4 Let $f_1, \ldots, f_N:X\to X$ be contractions in the complete metric space $X$, and $K$ the self-similar set with respect to the $f_i$. If $A\subset X$ is compact and $A\subset f_1(A)\cup\ldots\cup f_N(A)$, then $A\subset K$. Problem 5 Under the same hypothesis of the previous problem, if $B\subset X$ is nonempty and $B\supset f_1(B)\cup\ldots\cup f_N(B)$, then $\bar B \supset K$.

### Tarea 9, Análisis real

Due October 9 Problem 1 If $f:X\to Y$ is a Lipschitz function, then it is uniformly continuous. The function $x\to\sqrt x$ is uniformly continuous on $[0,\infty)$ but not Lipschitz. Problem 2 Let $P,Q,f:[-1,1]\to\R$ continuous and $a,b\in\R$. Then the IVP $\begin{cases} u''(x)+P(x)u'(x)+Q(x)u(x)=f(x)\\u(0)=a,\qquad u'(0)=b\end{cases}$ has a unique solution in a neighborhood of $x=0$. Problem 3 Consider the integral operator $\Phi:C([-1,1])\to C([-1,1])$ given by $\displaystyle\Phi(x)(t) = 1 + 2\int_0^t sx(s) ds,$ for $x\in C([-1,1])$. Starting from the constant function $x_0(t)=1$, explicitly calculate the iterations of $x_{n+1} = \Phi(x_n)$, and verify that $x_n$ is the $n$-th Taylor polynomial of $e^{t^2}$ around $t=0$. Problem 4 For $T\in L(\R^l,\R^m)$, $||T||_L = \sup\{ |Tx| : x\in\bar B_1(0)\}$. Problem 5 Let $B_r(x_0)$ be a ball in $\R^l$ and