Week 9-10 Problems, Stepan Malkov

Prove the parallelogram law: for any inner product space $V$ and $x,y \in V,$
$\|x+y\|^2 + \|x-y\|^2 = 2\|x\|^2 + 2\|y\|^2.$
Show that the space
$l^2(\mathbb{C}) = \{(x^n)_{n=1}^\infty, x_n \in \mathbb{C}: \langle (x^n), (x^m) \rangle = \sum_{i=1}^\infty x^{n}_i \overline{x^m_i} < \infty\}$
is a Hilbert space.
Give an example of an inner product space that is not a Hilbert space, i.e. an inner product space that is not complete.
If $f_n \in C(\mathbb{R}/\mathbb{Z})$ converges to $f$ in $L^2,$ does it converge uniformly? Prove the statement or find a counterexample.

Let $f(x,y) = \begin{cases} \frac{x^3}{x^2+y^2} & (x,y) \not = (0,0) \\ 0 & (x,y) = (0,0) \\ \end{cases}.$ Show that all partial derivatives of $f$ exist at $(0,0)$ but that $f$ is not differentiable at $(0,0).$ Explain why these two facts do not contradict each other.
Show that if $f \in C(\mathbb{R}/\mathbb{Z})$ is has a continuous derivative $g(x) = f'(x),$ then $\widehat{g}(n) = 2\pi i n\widehat{f}(n),$ in other words, the Fourier transform turns differentiation into multiplication.
For $f,g \in C(\mathbb{R}/\mathbb{Z}),$ define the convolution
$(f * g)(x) = \int_0^1 f(x-y) g(y) dy.$
Show that
$\widehat{f * g}(n) = \widehat{f}(n)\widehat{g}(n).$

Use the Fourier transform to solve the differential equation
$u''+4\pi^2 u=0,u(0)=u(1).$
(Hint: use problem 6).
Let $U$ be a neighborhood of $x \in \mathbb{R}^n$ and let $f: \mathbb{R}^n \to \mathbb{R}^m$ be continuous with continuous first order partial derivatives on $U.$ Show $f$ is differentiable at $x$ and $Df(x)$ is given by the Jacobian matrix.