Tarea 3, Análisis real

Due August 28th

Problem 1

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 $latex f\in\mathcal P_n$.


2. 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.


3. 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

$latex |r|_p = \dfrac{1}{p^\alpha}$.

1. $latex d_p(r,s) = |r-s|_p$ is a metric on $latex \Q$.
   Let $latex Q_p$ be the completion of $latex (\Q,d_p)$ (called the set of $latex p$-adic numbers).


2. The series $latex 1 + 2 + 4 + \ldots$ converges in $latex \Q_2$.


3. In general, the series $latex 1 + p + p^2 + \ldots$ converges in $latex \Q_p$.

Problem 3

1.  $latex (\R,d_B)$ is not totally bounded.


2. $latex B_M(0)$ is totally bounded in $latex \R^n$.

Problem 4

If $latex (X,d_1)$ and $latex (X,d_2)$ are homeomorphic, then $latex (X,d_1)$ is compact if and only if $latex (X,d_2)$ is compact.

Problem 5

If $latex X$ is a compact metric space, then it is complete.