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Mostrando las entradas de noviembre, 2015

### Tarea 16, Análisis real

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

### Tarea 13, Análisis real

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