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Mostrando las entradas de septiembre, 2015

Tarea 8, Análisis real

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

Tarea 7, Análisis real

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