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Mostrando las entradas de noviembre, 2015

Tarea 16, Análisis real

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.

Tarea 15, Análisis real

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(

Tarea 14, Análisis real

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.

Tarea 13, Análisis real

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