# Spring 2015

## 1
Let $(X, d)$ and $(Y, \rho)$ be metric spaces, $f: X\to Y$, and $x_0 \in X$.

Prove that the following statements are equivalent:

1. For every $\varepsilon > 0 \quad \exists \delta > 0$ such that $\rho( f(x), f(x_0)  ) < \varepsilon$ whenever $d(x, x_0) < \delta$.
2. The sequence $\theset{f(x_n)}_{n=1}^\infty \to f(x_0)$ for every sequence $\theset{x_n} \to x_0$ in $X$.

## 2
Let $f: \RR \to \CC$ be continuous with period 1. Prove that
$$
\lim _{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^{N} f(n \alpha)=\int_{0}^{1} f(t) d t \quad \forall \alpha \in \RR\setminus\QQ.
$$

> Hint: show this first for the functions $f(t) = e^{2\pi i k t}$ for $k\in \ZZ$.

## 3 

Let $\mu$ be a finite Borel measure on $\RR$ and $E \subset \RR$ Borel. 
Prove that the following statements are equivalent:

1. $\forall \varepsilon > 0$ there exists $G$ open and $F$ closed such that 
$$
F \subseteq E \subseteq G \quad \text{and} \quad \mu(G\setminus F) < \varepsilon.
$$
2. There exists a $V \in G_\delta$ and $H \in F_\sigma$ such that 
$$
H \subseteq E \subseteq V \quad \text{and}\quad \mu(V\setminus H) = 0
$$

## 4
Define
$$
f(x, y):=\left\{\begin{array}{ll}{\frac{x^{1 / 3}}{(1+x y)^{3 / 2}}} & {\text { if } 0 \leq x \leq y} \\ {0} & {\text { otherwise }}\end{array}\right.
$$

Carefully show that $f \in L^1(\RR^2)$.


## 5
Let $\mathcal H$ be a Hilbert space.

1. Let $x\in \mathcal H$ and $\theset{u_n}_{n=1}^N$ be an orthonormal set.
  Prove that the best approximation to $x$ in $\mathcal H$ by an element in $\spanof_\CC\theset{u_n}$ is given by
  $$
  \hat x \definedas \sum_{n=1}^N \inner{x}{u_n}u_n.
  $$
2. Conclude that finite dimensional subspaces of $\mathcal H$ are always closed.


## 6
Let $f \in L^1(\RR)$ and $g$ be a bounded measurable function on $\RR$.

1. Show that the convolution $f\ast g$ is well-defined, bounded, and uniformly continuous on $\RR$.
2. Prove that one further assumes that $g \in C^1(\RR)$ with bounded derivative, then $f\ast g \in C^1(\RR)$ and
$$
\frac{d}{d x}(f * g)=f *\left(\frac{d}{d x} g\right)
$$