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## Due August 28th

### Problem 1

1. For $n\in\Z_+$, let $\mathcal P_n$ the space of polynomials of degree at most $n$, seen as functions on $[0,1]$. If $f_n$ converges uniformly to $f$ on $[0,1]$, then $f\in\mathcal P_n$.

2. Consider the sequence
$f_n(x) = 1 + \dfrac{1}{2}x + \dfrac{1}{4}x^2 + \ldots + \dfrac{1}{2^n}x^n.$
Then $f_n$ converges uniformly in $C([0,1])$, but its limit is not a polynomial.

3. Let $\mathcal H$ be the subspace of $C([0,1])$ of functions satisfying $f(1-x) = f(x)$ for any $x\in[0,1]$ (these are called even function on $[0,1]$). Then $\mathcal H$ is an infinite dimensional closed subspace of $C([0,1])$.

### Problem 2

Let $p$ be a prime number. For $r\in\Q$, write $r = p^\alpha \dfrac{u}{v}$, where $\alpha,u,v\in\Z$ and $p$ does not divide neither of $u$ nor $v$. Define the function $|\cdot|_p:\Q\to\Q$ by

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

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

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

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

### Problem 3

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

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

### Problem 4

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

### Problem 5

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