Tarea 7, Análisis real

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 $latex C(X\times Y)$.