\documentclass{amsart}

\title{Problem Set the Last}

\begin{document}
\maketitle
The first few problems are from the book. I'll list them via number in the book plus the actual statement of the problem.

These are from Chapter 9.
\begin{enumerate}
\item[4.] Prove the parallelogram law.
\item[5.] Prove Appollonius' identity:
\[
||w-u||^2+||w-v||^2=\frac{1}{2}||u-v||^2+2\left\Vert w-\frac{1}{2}(u+v)\right\Vert^2.
\]
\item[7.] Prove that two vectors $u$ and $v$ in an real inner product space $V$ are orthogonal if and only if
\[
||u+v||^2=||u||^2+||v||^2.
\]
\item[8.] Show that an isometry is injective.
\item[15.] Let $V$ be a finite dimensional inner product space. Prove that for any subset $X$ of $V$, $X^{\perp\perp}=\langle X\rangle$.
\item[16.] Let $P_3$ be the inner product space of all polynomials of degree at most 3, under the inner product
\[
\langle p(x),q(x)\rangle=\int_{-\infty}^{\infty} p(x)q(x)e^{-x^2}dx.
\]
Apply Gram-Schmidt to the basis $\{1,x,x^2,x^3\}$.
\end{enumerate}

From Chapter 11
\begin{enumerate}
\item[2a.] Use the triangle inequality to prove
\[
|d(x,y)-d(a,b)|\leq d(x,a)+d(y,b).
\]
\item[2b.] Prove that
\[
|d(x,z)-d(y,z)|\leq d(x,y).
\]
\item[27.] Suppose that $(x_n)$ is a Cauchy sequence in a metric space $M$ and that some subsequence $(x_{n_k})$ of $(x_n)$ converges to $x$. Prove that $(x_n)$ converges to $x$.
\item[36.] Show that the metric spaces $C[a,b]$ and $C[c,d]$, under the sup metric, are isometric (hint: rescale).
\end{enumerate}

Now a fun one.

Use the Fourrier series for $x^3$ and Parseval's equality to show 
\[
\sum_{n\geq 1}\frac{1}{n^6}=\frac{\pi^6}{945}.
\]
\end{document}