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\title{ Math 5210, HW III \\ due April 20}
\maketitle
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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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\noindent
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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\noindent
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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