Ricardo A. Sáenz

Due November 27 Problem 1 If $latex f_n$ is a sequence of measurable functions, then $latex \{x:\lim f_n \text{ exists} \}$ is a measurable set. Problem 2 If $latex f:\R\to\R$ is monotone, then is Borel measurable. Problem 3 If $latex f_n\in L^+$ decreases pointwise to $latex f$, and $latex \int f_1 < \infty$, then $latex \int f = \lim \int f_n$. Problem 4 Let $latex f_n\in L^1$ such that $latex f_n\rightrightarrows f$. If $latex \mu(X)<\infty$, then $latex f\in L^1$ and $latex \int f_n\to\int f$. If $latex \mu(X)=\infty$, then the conclusions of (1.) might fail. Problem 5 If $latex 1\le p<r\le \infty$, $latex L^p\cap L^r$ is a Banach space with norm $latex ||f||=||f||_p + ||f||_r$. If $latex 1\le p<q<r\le\infty$, the inclusion map $latex L^p\cap L^r\to L^q$ is continuous.

Due November 20 Problem 1 $latex \mathcal B_\R$ is generated by each of the following: $latex \mathcal E_1 = \{(a,b): a,b\in\R, a<b\}$ $latex \mathcal E_2 = \{[a,b]: a,b\in\R, a<b\}$ $latex \mathcal E_3 = \{(a,b]: a,b\in\R, a<b\}$ $latex \mathcal E_4 = \{[a,b): a,b\in\R, a<b\}$ $latex \mathcal E_5 = \{(a,\infty): a\in\R\}$ $latex \mathcal E_6 = \{(-\infty,a): a\in\R\}$ $latex \mathcal E_7 = \{[a,\infty): a\in\R\}$ $latex \mathcal E_8 = \{(-\infty,a]: a\in\R\}$ Problem 2 An algebra $latex \sigma$-algebra iff it is closed under countable increasing unions. Problem 3 If $latex \mu_1, \ldots,\mu_n$ are measures on $latex (X,\mathcal M)$ and $latex a_1,\ldots,a_n\ge0$, then $latex \sum_{j=1}^n a_j\mu_j$ is a measure on $latex (X,\mathcal M)$. Problem 4 Let $latex (X,\mathcal M,\mu)$ be a measure space and $latex E_i\in\mathcal M$. $latex \mu(\liminf E_i) \le \liminf \mu(E_i)$. If $latex \mu(\cup_i E_i)<\infty$, then $latex \mu(\limsup E_i) \ge \limsup \mu(...

Due November 13 Problem 1 (Bessel inequality) Let $latex X$ be a separable Hilbert space and $latex \{v_k\}$ an orthonormal set (may be finite). Then $latex \displaystyle \sum_k |(x,v_k)|^2 \le ||x||^2.$ Problem 2 Consider the function $latex ||\cdot||_\infty:l^2\to[0,\infty)$ given by $latex ||(a_n)||_\infty = \sup\{|a_n|:n\ge 1\}$. $latex ||\cdot||_\infty$ is a norm in $latex l^2$ Is $latex ||\cdot||_\infty$ equivalent to the $latex l^2$-norm? Is $latex (l^2,||\cdot||_\infty)$ complete? Problem 3 Let $latex X$ be a separable Hilbert space and $latex Y$ a closed subspace. Then $latex Y$ is separable. Problem 4 Let $latex X$ be a separable inner product space and $latex \bar X$ its completion. Then $latex \bar X$ is a separable Hilbert space. Problem 5 Let $latex \phi\in X'$, where $latex X$ is a Hilbert space. Then $latex \ker\phi$ is a closed subspace of $latex X$ of codimension 1.

Due November 6 Problem 1 1. Let $latex X$ be a real inner product space. Then, for $latex x,y\in X$, $latex (x,y) = \dfrac{1}{4}\big( ||x+y||^2 - ||x-y||^2\big).$ 2. Let $latex X$ be a complex inner product space. The, for $latex x,y\in X$, $latex (x,y) = \dfrac{1}{4}\big( ||x+y||^2 - ||x-y||^2 + i||x+iy||^2 - i||x-iy||^2 \big).$ Problem 2 Let $latex (X,||\cdot||)$ be a normed space such that, for $latex x,y\in X$, $latex ||x + y||^2 +||x - y||^2 = 2||x||^2 + 2||y||^2.$ Then $latex ||\cdot||$ is induced by an inner product. Problem 3 Let $latex X$ be a finite dimensional inner product space over $latex \K$, say $latex \dim X = l$. Then $latex X$ is isometrically isomorphic to $latex \K^l$;  i.e. there exists an isomorphism $latex \Phi:X\to\K^l$ such that $latex (x,y) = (\Phi x, \Phi y)$ for all $latex x,y\in X$. Problem 4 Let $latex Y$ be a closed subspace of the Hilbert space $latex X$, and define $latex T:X\to Y$ as $latex Tx = \Proj_Y x$. Then $latex T$ is continuous. Problem 5 Let...

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 $latex V$ be a compact convex subset of the Banach space $latex X$ and $latex f:V\to V$ continuous. Then $latex f$ has a fixed point. Proof. For a given $latex n\in\Z_+$, let $latex x_1, \ldots, x_k\in V$ be such that $latex \displaystyle V\subset\bigcup_i^k B_{1/n}(x_i)$. Such $latex x_i$ exist because $latex V$ is compact. Define, for each $latex i$, the functions $latex \lambda_i:V\to\R$ by $latex \lambda_i(x) = \begin{cases}1/n-||x_i - x||& x\in B_{1/n}(x_i)\\0 & \text{otherwise.}\end{cases}$ The functions $latex \lambda_i$ are continuous and $latex \sum_i \lambda_i(x)\not=0$ for every $latex x\in V$. Thus, if we define $latex \pi_n(x) = \dfrac{\sum \lambda_i(x)x_i}{\sum\lambda_i(x)}$, $latex \pi_n$ maps $latex V$ into the ...

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

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

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

Due October 2nd Problem 1 For any subset $latex A$ of the metric space $latex X$, $latex \diam A = \diam \bar A$. Problem 2 Let $latex A\subset X$ be dense in $latex X$. If $latex E$ is closed in $latex X$ and $latex E\cap A = \emptyset$, then $latex E$ is nowhere dense. Problem 3 If $latex A\subset X$ is a $latex G_\delta$ set and dense in $latex X$, then $latex X\setminus A$ is of the first category. Problem 4 If $latex A$ and $latex X\setminus A$ are dense in the complete space $latex X$, then only one of them can be $latex F_\sigma$ in $latex X$. Problem 5 Let $latex A\subset X$ be countable and dense in the complete space $latex X$ without isolated points. Then $latex A$ is not a $latex G_\delta$ set.

Due September 25th Problem 1 Let $latex f,g\in C(X)$. Then the functions $latex \max(f,g):X\to\R$ and $latex \min(f,g):X\to\R$ are continuous. Problem 2 Let $latex a>0$. The space of continous even functions on $latex [-a,a]$ is a proper closed subalgebra of $latex C([-a,a])$. Problem 3 Let $latex f$ be a continuous function on $latex [a,b]$ such that $latex \displaystyle \int_a^b f(x) x^n dx = 0$ for all $latex n=0,1,2,\ldots$. Then $latex f=0$. Problem 4 Find a nontrivial continuous function $latex f:[0,1]\to\R$ such that $latex \displaystyle \int_0^1 f(x)dx = \int_0^1 f(x) x dx = \int_0^1 f(x)x^2dx = 0.$ Problem 5 Let $latex X,Y$ compact metric spaces, $latex X\times Y$ the product space and $latex \mathcal A$ the algebra generated by the functions $latex f:X\times Y\to \R$ of the form $latex f(x,y) = g(x)h(y), \qquad g\in C(X), \; h\in C(Y).$ Then $latex \mathcal A$ is dense in $latex C(X\times Y)$.

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 $la...

Due September 11th Problem 1 Let $latex f:X\to Y$ be a function, $latex A,B\subset X$ and $latex U,V\subset Y$. $latex f(A\cup B) = f(A)\cup f(B)$. $latex f(A\cap B)\subset f(A)\cap f(B)$. Give an example where $latex f(A\cap B)\not\supset f(A)\cap f(B)$. $latex f^{-1}(U\cup V) = f^{-1}(U)\cup f^{-1}(V)$. $latex f^{-1}(U\cap V) = f^{-1}(U)\cap f^{-1}(V)$. $latex f(f^{-1}(U)) \subset U$. Give an example where $latex f(f^{-1}(U))\not\supset U$. $latex f^{-1}(f(A)) \supset A$. Give an example where $latex f^{-1}(f(A))\not\subset A$. Problem 2 If $latex X$ is sequentially compact and $latex f:X\to Y$ is continuous, then $latex f(X)$ is sequentially compact. Prove it directly using the definition of sequential compactness. Problem 3 Give a set $latex X$ and two metrics $latex d,d'$ on $latex X$ such that $latex (X,d)$ and $latex (X,d')$ are homeomorphic, but $latex f:X\to X$ given by $latex f(x)=x$ is not uniformly continuous. Problem 4 1.  Let $latex \mathcal I:C([0,1]\...

Due September 4th Problem 1 Let $latex X$ be sequentially compact. Then every infinite subset of $latex X$ has a limit point. (Do not use the Bolzano-Weierstrass theorem.) Problem 2 Let $latex E$ be a compact subset of $latex \R$. Then it has a minimum and a maximum. Problem 3 Let $latex A$ be a bounded infinite subset of $latex \R^l$. Then it has a limit point. Problem 4 Let $latex (x_n)$ be a bounded sequence in $latex \R^l$. Then is has a convergent subsequence. Problem 5 Let $latex A$ be a nonempty set in the metric space $latex (X,d)$ and, for $latex \e>0$, define $latex A_\e = \{x\in X: d(x,A) < \e\}.$ Then $latex A_\e$ is open in $latex X$.

Due August 28th Problem 1 For $latex n\in\Z_+$, let $latex \mathcal P_n$ the space of polynomials of degree at most $latex n$, seen as functions on $latex [0,1]$. If $latex f_n$ converges uniformly to $latex f$ on $latex [0,1]$, then $latex f\in\mathcal P_n$. Consider the sequence $latex f_n(x) = 1 + \dfrac{1}{2}x + \dfrac{1}{4}x^2 + \ldots + \dfrac{1}{2^n}x^n.$ Then $latex f_n$ converges uniformly in $latex C([0,1])$, but its limit is not a polynomial. Let $latex \mathcal H$ be the subspace of $latex C([0,1])$ of functions satisfying $latex f(1-x) = f(x)$ for any $latex x\in[0,1]$ (these are called  even function on  $latex [0,1]$). Then $latex \mathcal H$ is an infinite dimensional closed subspace of $latex C([0,1])$. Problem 2 Let $latex p$ be a prime number. For $latex r\in\Q$, write $latex r = p^\alpha \dfrac{u}{v}$, where $latex \alpha,u,v\in\Z$ and $latex p$ does not divide neither of $latex u$ nor $latex v$. Define the function $latex |\cdot|_p:\Q\to\Q$ by $latex |r|_p = ...

Due August 21 Problem 1 Let $latex (X,||\cdot||)$ be a normed vector space and $latex x_n, y_n$ sequences in $latex X$ such that $latex x_n\to x$ and $latex y_n \to y$. Then $latex \lambda x_n + \mu y_n \to \lambda x + \mu y$ for any $latex \lambda,\mu\in\mathbb K$. Problem 2 If $latex (X,d)$ and $latex (X,d')$ be homeomorphic metric spaces, then they have the same convergence sequences. However, there exists homeomorphic metric spaces $latex (X,d), (X,d')$ such that only one of them is complete. Problem 3 If $latex (X,||\cdot||)$ and $latex (X,||\cdot||')$ are homeomorphic, then $latex (X,||\cdot||)$ is complete if and only if $latex (X,||\cdot||')$ is complete. Problem 4 Let $latex  f_n$ be the sequence of functions in $latex C([0,1])$ given by $latex \displaystyle f_n(x) = \begin{cases}\sqrt n & 0\le x<\dfrac{1}{n}\\\dfrac{1}{\sqrt x} & \dfrac{1}{n}\le x\le 1.\end{cases}$ Then $latex f_n$ is a Cauchy sequence in $latex (C([0,1]),||\cdot||_1)$ that do...

Due August 14 Problem 1 The function $latex d_T(x,y) = |x^1 - y^1| + \ldots + |x^n - y^n|$ defined for $latex x,y\in\R^n$ is a metric on $latex \R^n$. Problem 2 Two norms $latex ||\cdot||_1$ and $latex ||\cdot||_2$ on a vector space are  equivalent if there exist constants $latex c_1, c_2$ such that $latex c_1||x||_1 \le ||x||_2 \le c_2||x||_1$ for all $latex x\in X$. The norms $latex ||\cdot||_E, ||\cdot||_M$ and $latex ||\cdot||_T$ on $latex \R^n$ are equivalent. If $latex ||\cdot||_1$ and $latex ||\cdot||_2$ are equivalent and $latex B^i_r(x)$ is the ball of radius $latex r$ with center $latex x$ with respect to the metric induced by $latex ||\cdot||_i$, then, for each $latex \e>0$, there exist $latex \delta_1,\delta_2>0$ such that $latex B^1_{\delta_1}(x) \subset B_\e^2(x)$ and $latex B^2_{\delta_2}(x)\subset B_\e^1(x)$. Let $latex ||\cdot||_1$ and $latex ||\cdot||_2$ two norms on $latex X$, and suppose there exist $latex \delta, \e>0$ such that $latex B^1_\delta(0)...

Los exámenes extraordinarios de las materias Análisis real y Análisis de Fourier se llevarán a cabo el miércoles 13 de enero, 3:00pm.

Problema 2. Existe $latex \e>0$ tal que el PVI $latex \displaystyle \begin{cases}x'(t)=\sin x - tx\\x(0)=1\end{cases}$ tiene solución en $latex [-\e,\e]$.

Pueden hacer aquí preguntas y comentarios de las clases Análisis real y Análisis de Fourier , con miras al examen de este viernes.

Aquí está el capítulo final de las notas de clase: Espacios de Hilbert . El archivo completo con las notas del semestre esta aquí: Análisis real: Primer curso . Estas notas siguen en revisión, pero pueden ser consideradas como "finales" para propósitos de este curso. (El capítulo 8, por ejemplo, que no fue cubierto en clase, no está terminado.)