# Fall 2014

## 1
Let $\theset{f_n}$ be a sequence of continuous functions such that $\sum f_n$ converges uniformly.

Prove that $\sum f_n$ is also continuous.


## 2
Let $I$ be an index set and $\alpha: I \to (0, \infty)$.

1. Show that
$$
\sum_{i \in I} a(i):=\sup _{\substack{ J \subset I \\ J \text { finite }}} \sum_{i \in J} a(i)<\infty \implies I \text{ is countable.}
$$

2. Suppose $I = \QQ$ and $\sum_{q \in \mathbb{Q}} a(q)<\infty$.
  Define
  $$
  f(x):=\sum_{\substack{q \in \mathbb{Q}\\ q \leq x}} a(q).
  $$
  Show that $f$ is continuous at $x \iff x\not\in \QQ$.

## 3
Let $f\in L^1(\RR)$. Show that
$$
\forall\varepsilon > 0 ~~\exists \delta > 0 \text{ such that } m(E) < \delta \implies \int_{E}|f(x)| d x<\varepsilon
$$

## 4
Let $g\in L^\infty([0, 1])$
Prove that
$$
\int_{[0,1]} f(x) g(x) d x=0 \quad\text{for all continuous } f:[0, 1] \to \RR \implies g(x) = 0 \text{ almost everywhere. }
$$

## 5

1. Let $f \in C_c^0(\RR^n)$, and show
$$
\lim _{t \rightarrow 0} \int_{\mathbb{R}^{n}}|f(x+t)-f(x)| d x=0.
$$

2. Extend the above result to $f\in L^1(\RR^n)$ and show that
$$
f\in L^1(\RR^n),~ g\in L^\infty(\RR^n) \implies f \ast g \text{ is bounded and uniformly continuous. }
$$

## 6
Let $1 \leq p,q \leq \infty$ be conjugate exponents, and show that
$$
f \in L^p(\RR^n) \implies \|f\|_{p}=\sup _{\|g\|_{q}=1}\left|\int f(x) g(x) d x\right|
$$