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)...

Fecha de entrega: 29 de mayo Problema 1 Sea $latex V$ un espacio vectorial de dimensión $latex n$, $latex u\in V$ y $latex W = \{u\}^\perp$. Muestra que, si $latex \{u_1, \ldots, u_{n-1} \}$ y $latex \{v_1, \ldots, v_{n-1} \}$ son dos bases de $latex W$ tales que $latex [u, u_1, \ldots, u_{n-1} ] = [ u, v_1, \ldots, v_{n-1} ]$, entonces $latex [u_1, \ldots, u_{n-1} ] = [ v_1, \ldots, v_{n-1} ]$. ( Sugerencia: Escribe cada $latex v_i$ en la base $latex \{u_j\}$ y muestra que el determinante de la matriz de cambio de base es positivo.) Problema 2 Sea $latex M^{n-1}\subset\R^n$ una variedad diferenciable con orientación $latex \mu$. Para cada $latex p\in M$, definimos el vector normal $latex \nu_p\in(M_p)^\perp\subset\R^n$ como el vector unitario tal que $latex [\nu_p, (v_1)_p, \ldots, (v_{n-1})_p]$ es la orientación estándar de $latex \R^n$, para cualquier base $latex \{(v_1)_p, \ldots, (v_{n-1})_p\}$ de $latex M_p$ tal que $latex [(v_1)_p, \ldots, (v_{n-1})_p] = \mu_p$. Muestra que el ...

Fecha de entrega: 22 de mayo Problema 1 Describe el espacio tangente del cilindro $latex C = \{(x,y,z)\in\R^3: x^2 + y^2 = 1, 0\le z\le 1\}$ en un punto $latex p = (\cos\theta, \sen\theta, z_0)$, $latex \theta\in[0,2\pi), z_0\in[0,1]$. Problema 2 Verifica que las siguientes funciones en $latex C$ son campos vectoriales del cilindro. $latex \displaystyle F(x,y,z) = \begin{pmatrix} y \\ -x \\ 0 \end{pmatrix}_{(x,y,z)}$ $latex \displaystyle F(x,y,z) = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}_{(x,y,z)}$ $latex \displaystyle F(x,y,z) = \begin{pmatrix} yz \\ -xz \\ z^2 \end{pmatrix}_{(x,y,z)}$ $latex \displaystyle F(x,y,z) = \begin{pmatrix} xy - yz \\ xz - x^2 \\ yz - z^2 \end{pmatrix}_{(x,y,z)}$ Problema 3 Considera la 1-forma en el cilindro $latex \omega(p) = z_0\cos^2\theta dx - \cos\theta\sen\theta dy + z_0\sen2\theta dz,$ para $latex p = (\cos\theta, \sen\theta, z_0)$, $latex \theta\in[0,2\pi), z_0\in[0,1]$. Utiliza un sistema de coordenadas apropiado para calcular $latex d\ome...

Fecha de entrega: 15 de mayo Problema 1 Demuestra que un subespacio de dimensión $latex k$ de $latex \R^n$ es una variedad diferenciable de dimensión $latex k$. Problema 2 Sea $latex g:\R^2\to\R$ dada por $latex g(x,y) = x^2 - y^2$. Explica por qué el conjunto $latex g^{-1}(\{0\})$ no es una variedad diferenciable en dimensión 1 en $latex \R^2$. Problema 3 Sea $latex A\subset\R^n$ un conjunto abierto y $latex g:A\to\R$ continuamente diferenciable tal que $latex g'(x)\not=0$ para $latex x\in A$. Si $latex M = g^{-1}(\{0\})\not=\emptyset$, muestra que el espacio tangente en $latex x\in M$ es igual a $latex \{v_x\in\R^n_x: v \cdot \grad g(x) = 0 \}$. Es decir, $latex M_x$ es el hiperplano en $latex \R^n_x$ ortogonal a $latex \grad g(x)$, el gradiente de $latex g$ en $latex x$. Problema 4 Sea $latex f:\R^n\to\R^m$ y considera su gráfica $latex G = \{(x,y)\in\R^{n+m}: y = f(x)\}$. Muestra que $latex G$ es una variedad diferenciable de dimensión $latex n$ si, y solo si, $latex f$ es de...

Fecha de entrega: 8 de mayo Problema 1 Muestra que el anillo $latex \mathbb A = \{(x,y)\in\R^2: 1\le x^2 + y^2 \le 2\}$ es un 2-cubo. Calcula $latex \partial\mathbb A$. Problema 2 Muestra que un conjunto abierto simplemente conexo es conexo. Problema 3 Sea $latex \w$ la 1-forma en $latex \R^2\setminus\{0\}$ dada por $latex \displaystyle \w = - \frac{y}{x^2 + y^2} dx + \frac{x}{x^2 + y^2} dy$. Muestra que, si $latex n\in\Z$ y $latex s_{R,n} (t) = (R \cos 2\pi nt, R \sen2\pi nt)$, entonces $latex \displaystyle \int_{s_{R,n}} \w = 2\pi n.$ Problema 4 Sea $latex \w = yzdx + xzdy + xydz$. Calcula $latex \displaystyle\int_c \w$ para las siguientes curvas. $latex c(t) = (\cos 2\pi t, \sen 2\pi t, \sen \pi t)$; $latex c(t) = (t, t^2, t^3)$; $latex c(t) = (t, 2t^2 - t, t)$. Problema 5 Sea $latex \w = x dy\wedge dz + y dz\wedge dx + zdx\wedge dy$ y $latex S$ la superficie dada por la gráfica de la función $latex f(x,y) = x^2 + y^2$ en $latex [-1,1]\times[-1,1]$. Calcula $latex \displaystyl...

Fecha de entrega: 1 de mayo Problema 1 Sea $latex f:\R^n\to\R$ diferenciable tal que $latex \grad f(p)\not=0$. Muestra que $latex \grad f(p)$ es el vector con la dirección de crecimiento más rápido de $latex f$ en el punto $latex p$. Es decir, si $latex \hat u = \dfrac{\grad f(p)}{|\grad f(p)|}$, entonces $latex Df(p)(\hat u) = \max\{ Df(p)(v): |v|=1\}$. ( Sugerencia:  Nota que $latex (\grad f(p))_p\cdot v_p = Df(p)(v)$, para $latex v_p\in\R^n_p$.) Problema 2 Sea $latex \w = f dx$ una 1-forma en $latex [0,1]$ tal que $latex f(0) = f(1)$. Muestra que existe un único $latex \lambda\in\R$ tal que $latex \w - \lambda dx = dg$, donde $latex g$ es una función que satisface $latex g(0) = g(1)$. Problema 3 Sea $latex \w = \w_1 dx + \w_2 dy + \w_3 dz$ una 1-forma diferencial tal que $latex \w_1,\w_2,\w_3$ son homogéneas de grado $latex \alpha$. Muestra que, si $latex \w$ es cerrada, entonces $latex \w = df$ donde $latex f(x,y,z) = \dfrac{1}{\alpha+1}(\w_1(x,y,z)x + \w_2(x,y,z)y + \w_3(x,y,z)z)$...

Fecha de entrega: 24 de abril Problema 1 Calcula el producto cuña $Latex \phi \wedge \psi$ de las siguientes 1-formas en $Latex \mathbb R^3$. a) $Latex \phi = 3dx+dz, \quad \psi=dy-dz$; b) $Latex \phi=dx-dy+2dz,\quad \psi=3dx-4dy-2dz$. Escribe el resultado en la base $Latex dy\wedge dz, dz\wedge dx, dx\wedge dy$. Problema 2 Calcula el diferencial $Latex d\omega$ de las siguientes 1-formas diferenciales en $Latex \mathbb R^3$ a) $Latex \omega(x,y,z)=(z^2-x ^2)dx+(y^2-z^2)dy+(x^2-y^2)dz$; b) $Latex \omega(x,y,z)=(3x^2-y^2z)dx-2xyzdy-xy^2dz$. Problema 3 Calcula $Latex \omega\wedge\eta$, para las siguientes formas diferenciales en $Latex \mathbb R^3$. a) $Latex \omega=xdx-ydy,\quad \eta=zdz\wedge dy+xdy\wedge dz$; b) $Latex \omega=dx+dy+dz,\quad \eta=dx\wedge dy+dx\wedge dz+dy\wedge dz$; c) $Latex \omega=zdx\wedge dy+xdy\wedge dz,\quad \eta=\omega$. Problema 4 Sea $Latex \omega$ la 2-forma diferencial en $LAtex \mathbb R^{2n}$ dada por $Latex \displaystyle \omega=dx^1\wedge dx^2+dx^3\wedge...

Fecha de entrega: 17 de abril Problema 1 Muestra que la integral impropia $Latex \displaystyle \int_{\mathbb R^n} e^{-|x|^2} dx = \lim_{N\rightarrow\infty}\int_{[-N,N]^n} e^{-|x|^2}dx = \pi^{n/2}.$ Problema 2 Si $Latex x>0$, muestra que la función $LAtex t\mapsto t^{x-1}e^{-t}$ es integrable en $Latex \mathbb R_+ = (0,\infty).$ Para $Latex x>0$, la función gama $Latex \Gamma(x)$ está definida como la integral $Latex \displaystyle \Gamma(x) = \int_{\mathbb R_+}t^{x-1}e^{-t} dt$. Muestra que $Latex \displaystyle \Gamma(x) = \lim_{\varepsilon\rightarrow 0} \int_\varepsilon ^{1/\varepsilon} t^{x-1} e^{-t}.$ Muestra que, si $Latex n\in \mathbb Z_+$, entonces $Latex \displaystyle \Gamma(n+1) = n\Gamma(n).$ Concluye que, para $Latex n\in\mathbb N,\;\;n!=\Gamma(n+1).$ Problema 3 Muestra que el determinante del jacobiano de las coordenadas esféricas está dado por $Latex \displaystyle r^{n-1}\sin^{n-2}\phi_1\sin^{n-3}\phi_2\cdots\sin\phi_{n-2}$, para cada $Latex r>0$, $latex 0...

Fecha de entrega: 27 de marzo Problema 1 Sean $latex a < b\in\R$. Muestra que existe $latex f\in C^\infty(\R)$ tal que $latex f > 0$ en $latex (a,b)$ y $latex f(x) = 0$ para $latex x\notin(a,b)$. Sean $latex a < b\in\R$. Muestra que existe $latex f\in C^\infty(\R)$ tal que $latex 0\le f\le 1$, $latex f(x)=0$ para $latex x\le a$ y $latex f(x)=1$ para $latex x\ge b$. Sean $latex R>r>0$. Muestra que existe $latex f\in C^\infty(\R^n)$ tal que $latex f=1$ en $latex B_r(0)$ y $latex \supp f = B_R(0)$. Problema 2 Sean $latex C,E\subset\R^n$ tales que $latex C$ es compacto, $latex E$ es cerrado y $latex C\cap E = \emptyset$. Muestra que existe un conjunto compacto $latex D\subset\R^n$ tal que $latex C\subset D^0$ y $latex D \cap E = \emptyset$. Problema 3 Sean $latex C,E\subset\R^n$ tales que $latex C$ es compacto, $latex E$ es cerrado y $latex C\cap E = \emptyset$. Muestra que existe $latex f\in C^\infty(\R^n)$ tal que $latex f = 1$ en $latex C$ y $latex f = 0$ en $latex ...

Fecha de entrega: 20 de marzo Problema 1 Sea $latex f:R\to\R$ y $latex \mathcal P$ una partición de $latex R$. Muestra que $latex f$ es Riemann-integrable si y solo si $latex f|_S$ es Riemann-integrable para cada $latex S\in\mathcal P$, y en tal caso $latex \displaystyle \int_R f = \sum_{S\in\mathcal P} \int_S f|_S.$ Problema 2 Muestra que un conjunto no acotado no puede ser de contenido 0. Da un ejemplo de un conjunto cerrado de medida 0 que no sea de contenido 0. Si $latex C$ es de contenido 0, muestra que $latex \fr C$ es de contenido 0. Sin embargo, da un ejemplo de un conjunto de medida 0 cuya frontera no sea de medida 0. Problema 3 Sea $latex f:R\to\R$ Riemann-integrable, $latex f\ge 0$ y tal que $latex \int f = 0$. Muestra que $latex \{x\in R:f(x)\not=0\}$ es de medida 0. Problema 4 Muestra que, si $latex C$ es de contenido 0, entonces es Jordan-medible. Muestra que, si $latex C$ es Jordan-medible y de medida 0, entonces $latex \int_C 1 = 0$. Problema 5 Sea $latex f:...

Fecha de entrega: 13 de marzo Problema 1 Sea $latex K$ un polítopo convexo. Muestra que $latex K$ tiene un número finito de puntos extremos. Problema 2 Muestra que un polítopo compacto es la unión finita de simplejos. Si el polítopo tiene $latex r$ vectores linealmente independientes, muestra que es la unión finita de $latex r$-simplejos. Problema 3 Sea $latex f:R\to\R$ Riemann-integrable y $latex c\in\R$. Muestra que $latex cf$ es Riemann-integrable y $latex \displaystyle\int cf = c\int f$. Problema 4 Sean $latex f,g:R\to\R$ Riemann-integrables tales que $latex f\le g$. Muestra que $latex \displaystyle\int f \le \int g$. Problema 5 Sea $latex f:[a,b]\to\R$ creciente. Si $latex x_1,\ldots,x_k\in[a,b]$ son distintos, muestra que $latex \displaystyle\sum_{i=1}^k O(f,x_i) < f(b) - f(a).$

Fecha de entrega: 6 de marzo Problema 1 Sea $Latex f:\mathbb{R}^2\to\mathbb{R}$ continuamente diferenciable. Muestra que $Latex f$ no es inyectiva. ( Sugerencia: Considera la función $Latex g(x,y) = (f(x,y),y).$) Generaliza este resultado a funciones continuamente diferenciables $Latex f:\mathbb{R}^n\to\mathbb{R}^m,$ con $Latex m<n$. Problema 2 Muestra que si $Latex f:\mathbb{R}\to\mathbb{R}$ satisface $Latex f'(x)\neq 0$ para todo $Latex x\in\mathbb{R}$, entonces $Latex f$ es inyectiva. Sin embargo, muestra que $Latex f:\mathbb{R}^2\to\mathbb{R}^2$ dada por  $Latex \displaystyle f(x,y) = (e^x\cos{y},e^x\sin{y})$ satisface $Latex \det f'(x,y)\neq 0$ para todo $Latex (x,y)\in\mathbb{R}^2$, pero no es inyectiva. Problema 3 Sea $Latex K\subset\mathbb{R}^n$ un conjunto convexo cerrado no vacío tal que $Latex \mathbb{R}^n\setminus K\neq\emptyset$ es convexo. Muestra que $Latex K$ es un semiespacio cerrado. Problema 4 Sean $Latex f,g:K\to\mathbb{R}$ convexas y sea $Latex ...

Fecha de entrega: 27 de febrero Problema 1 Si extendemos la definición de derivadas direccionales a vectores $latex u$ no necesariamente unitarios, demuestra que satisfacen $latex D_{tu}f(x_0) = t D_uf(x_0)$ y $latex D_{u+v}f(x_0) = D_u f(x_0) + D_v f(x_0),$ si $latex f$ es diferenciable en $latex x_0$. Problema 2 Si $latex f:U\to\R$ tiene un mínimo local en $latex x_0$ y sus derivadas parciales existen, muestra que $latex D_if(x_o)=0$ para cada $latex i=1,\ldots,n$. Problema 3 Muestra que, si $latex U\subset\R^n$ es abierto, $latex f:U \to \R$ es tal que sus derivadas parciales existen en cada $latex x\in U$, $latex x_0\in U$, y $latex t\in\R$ es tal que $latex (x_0^1, \ldots, x_0^i + s, \ldots, x_0^n) \in U$ para todo $latex s\in[0,t]$ (o $latex s\in[t,0]$, si $latex t<0$), entonces existe $latex c$ entre $latex x_0^i$ y $latex x_0^i+t$ tal que $latex f(x_0^1,\ldots,x_0^i+t,\ldots,x_0^n) - f(x_0^1,\ldots,x_0^i,\ldots,x_0^n) = t D_if(x_0^1,\ldots,c,\ldots,x_0^n).$ Problema 4 Sea $l...

Fecha de entrega: 20 de febrero Problema 1 Si $latex f:U\to\R^m$ es diferenciable en $latex x_0\in U$, entonces es continua en $latex x_0$. Problema 2 Sea $latex U\subset\R^n$ abierto y $latex f,g:U\to\R$ tales que $latex f$ es continua en $latex x_0\in U$, $latex g$ es diferenciable en $latex x_0$ y $latex g(x_0) = 0$. Muestra que $latex fg$ es diferenciable en $latex x_0$. Problema 3 Calcula la derivada y el Jacobiano de cada una de las siguientes funciones, utilizando la regla de la cadena. $latex (x,y) \mapsto (x^2 - y^2, 2xy)$, en cada punto $latex (x_0, y_0)\in\R^2$ $latex (x,y) \mapsto (\sen(x^2 + xy + y^2), e^{xy} )$, en cada punto $latex (x_0,y_0)\in\R^2$ Problema 4 Decimos que $latex f:\R^n\to\R$ es homogénea de grado $latex \alpha$ si $latex f(tx) = t^\alpha f(x)$, para $latex x\in\R^n, t>0$. Si, además, $latex f$ es diferenciable, muestra la fórmula de Euler $latex \displaystyle\sum_{i=1}^n x^i D_if(x) = \alpha f(x)$. Problema 5 Si $latex f:\R^n\to\R$ es diferenciab...

Fecha de entrega: 13 de febrero Problema 1 Muestra que, si $latex f:A\to\R^m$ tiene límites $latex L$ y $latex M$ en $latex x_0$, entonces $latex L = M$. Problema 2 Demuestra que la función $latex f:A\to\R^m$ es continua en $latex x\in A$ si y solo si cada una de sus componentes $latex f^i:A\to\R$ es continua en $latex x$. Problema 3 Considera la función en $latex \R^2$ definida por $latex f(x,y) = \begin{cases}\dfrac{xy}{x^2 + y^2} & (x,y)\not=(0,0)\\0 & (x,y)=(0,0).\end{cases}$ Muestra que, aunque cada una de las funciones $latex x\to f(x,y_0)$    y    $latex y\to f(x_0,y)$ son continuas en $latex \R$ para cualquier $latex x_0,y_0\in\R$, la función $latex f$ no es continua en $latex (0,0)$. Problema 4 Da un ejemplo de un conjunto $latex A\subset\R^n$ no acotado tal que toda función continua en $latex A$ es uniformemente continua. Problema 5 Calcula la oscilación en el punto $latex (0,0)$ de la función del problema 3.

Fecha de entrega: 6 de febrero Problema 1 Sea $latex (x_k)$ una sucesión en $latex \R^n$ tal que  $latex x_k\to L$ y $latex x_k\to M$. Muestra que $latex L = M$. Problema 2 Si $latex (x_k)$ es una sucesión de Cauchy, entonces es acotada. Sea $latex (x_k)$ una sucesión de Cauchy tal que una subsucesión converge, digamos $latex x_{k_l} \to L$. Muestra que $latex x_k\to L$. Concluye que toda sucesión de Cauchy en $latex \R^n$ converge. (Utiliza el teorema de Bolzano-Weierstrass.) Problema 3 Considera, en $latex \R^n$, la cubierta $latex \{A_n\}_n$ definida por $latex A_n = \{ x\in\R^n: \dfrac{1}{2n} < |x| < \dfrac{3}{2n}\},$ para la bola punteada $latex B_1^*(x) = \{ x: 0 < |x| \le 1 \}$. Muestra que esta cubierta no tiene subcubiertas finitas. Problema 4 Sean $latex A_1\supset A_2\supset\ldots$ compactos no vacíos en $latex \R^n$. Muestra que $latex \bigcap_i A_i \not=\emptyset.$ Muestra que el enunciado anterior es falso si los $latex A_i$ son solo cerrados. Problema ...

Fecha de entrega: 30 de enero Problema 1 Muestra la desigualdad del triángulo inversa: si $latex x,y\in\R^n$, $latex \big| |x| - |y|\big| \le |x-y|.$ Demuestra la identidad del palalelogramo: si $latex x,y\in\R^n$, $latex |x|^2 + |y|^2 = \frac{1}{2}\big(|x+y|^2 + |x-y|^2 \big).$ Problema 2 Muestra que, si $latex x_1,x_2\in\R^n$, el conjunto $latex \{x\in\R^n: |x - x_1| = |x - x_2| \}$ es un hiperplano. Problema 3 Muestra que la intersección de dos rectángulos en $latex \R^n$ es vacía o es otro rectángulo. Problema 4 Muestra que si $latex \{U_\alpha\}$ es una colección de conjuntos abiertos en $latex \R^n$, entonces la unión $latex \bigcup_\alpha U_\alpha$ es un conjunto abierto. Muestra que si $latex U_1,U_2,\ldots,U_k$ son conjuntos abiertos en $latex \R^n$, entonces la intersección $latex \bigcap_{i=1}^k U_i$ es un conjunto abierto. Problema 5 Muestra que, para cualquier $latex A\subset\R^n$, $latex \fr A = \bar A \cap \overline{(\R^n \setminus A)}$.