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 \title{ Math 5210,  HW III \\ due April 20}
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\noindent 
1)  Let $(Y,d)$ be a complete metric space and $X$ a dense subset of $Y$. The set $X$ is a also a metric space with respect to the same metric. Let $X^*$ be the 
completion of $X$. Recall that $X^*$ is the set of equivalence classes of Cauchy sequences $(x_n)$ in $X$. Since $Y$ is complete, $\lim_n x_n$ exists in $Y$. 
Equivalent Cauchy sequences have the same limit, hence $f((x_n))=\lim_n x_n$ is a well defined map $f: X^*\rightarrow Y$. Show that $f$ is an isomorphism of 
metric spaces. 

 
 
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\noindent 
2)  Let $V=C([0,1])$ be the space of continuous functions on $[0,1]$. Prove that the set of piece-wise linear function (i.e. whose graphs are obtained 
by connecting the dots in the plane) is dense in $V$, with respect to the sup norm, that is, for every $f\in V$  and every $\epsilon >0$, 
there exists a piece-wise linear function $g$ such that $|f(x)-g(x)| <\epsilon $ for all $x\in [0,1]$.  Hint: use uniform continuity of $f$. 

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\noindent 
3) Fix $K(x,y)$, a continuous function on $[0,1]^2$. Let $f(x)$ be a continuous function on $[0,1]$. Let 
\[ 
g(x)=\int_0^1 K(x,y) f(y) ~dy. 
\] 
Prove that $g(x)$ is a continuous function on $[0,1]$. Hint: $K$ is uniformly continuous, why? Let $V=C([0,1])$ be the space of continuous functions on $[0,1]$.  
Consider $V$ as a normed space with the sup norm. Let $T: V \rightarrow V$,  $T(f)=g$ for every $f\in V$, as above. Prove that $T$ is bounded. 

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\noindent 
4)
Let $U$ be a  dense subspace of a normed space $V$. Let $g: U\rightarrow  \mathbb R$ be a bounded linear functional i.e. there 
exists $C\geq 0$ such that 
\[ 
|g(x)| \leq C ||x|| 
\] 
for all $x\in U$. Then $g$ can be extended (uniquely) to a  linear functional $f: V \rightarrow \mathbb R$ satisfying the same bound.  
Hint: any $x\in V$ is a limit of a Cauchy sequence $(x_n)$ in $U$. 

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5) Recall the normed space $\ell^2(\mathbb N)$,  the set of all infinite tuples of real numbers  $x= (x_1, x_2, \ldots )$ such that 
$||x||^2 = \sum_{i=1}^{\infty} x_i^2  < \infty $, with the norm $||x||$ so defined. Let $S\subset \ell^2(\mathbb N)$ be the 
subset of all $x$ with $x_i\in \mathbb Q$ and almost all $x_i=0$. This is a countable set. Prove that 
$S$ is dense. 

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\noindent 
6) Let $V$ be a normed space, and $A, B\subset V$ two open sets. Prove that 
\[ 
A+B =\{ x+y ~|~ x\in A, y\in B\} 
\] 
is open. 

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\noindent 
7)  Perhaps you have seen the formula 
\[ 
\sum_{n=1}^{\infty} \frac{1}{n^2} =\frac{\pi^2}{6}. 
\] 
Where does this come from? The purpose of this exercise is to derive this formula as a special case of the Parseval's identity. Let $X=(-1/2,1/2]$. 
Let $f(x)=x$ on $X$. Compute $||f||^2$, the square of $L^2(X)$ norm of $f$. Then Fourier expand $f$ and then compute $||f||^2$ using the Parseval's 
identity.  (Be careful, the norm of $\sin(2\pi nx)$ is not 1). Deduce the identity. 

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8) Let $M\geq 0$. Let $c_n$ be a sequence of real numbers such that $|c_n| \leq M/n^2$ for all $n$. Then the series 
\[
f(t)=\sum_{n=1}^{\infty} c_n \sin(2\pi nt) 
\] 
converges uniformly, for all $t\in \mathbb R$. Hence $f$ is a periodic and continuous function $f$.  Prove that 
the series converges to $f$ in $L^2((-1/2,1/2])$ that is 
\[ 
\lim_n ||f-f_n|| = 0
\] 
where $f_n$ is the sequence of partial sums, and $||\cdot ||$ the $L^2$-norm. Hint: use Lebesgue dominated convergence theorem. 

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\noindent 
9) Let $V$ be a Hilbert space. Let $W\subset V$ be a closed subspace. Prove that $W$ contains a dense countable set, so it is also a Hilbert space. 
Hint: consider the projection $P: V \rightarrow W$. 

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