Anonymous Quiz for 18.155
September 6, 2001

Richard Melrose

Department of Mathematics, Massachusetts Institute of Technology

rbm@math.mit.edu

This anonymous quiz is just to help me judge the level at which I should begin. Please put an A, B or C in the margin on the left next to each question, where

1. [A] Means that you know the answer straight away or how to prove the statement.
2. [B] Means that you believe you could work it out in five or ten minutes.
3. [C] Means that you suspect you don't know some necessary underlying results or do not understand the statement.

Note that there are several statements here that I expect you not to know or understand.

1. Let $\mathcal{C}_0([-N,N])$ be the space of continuous functions on $\bbR$ which vanish outside $[-N,N]\subset\bbR.$ Let $\mathcal{C}_{\infty}(\bbR)$ be the space of bounded continuous functions on $\bbR$ with the supremum norm. Is the union $\bigcup_N\mathcal{C}_0([-N,N])$ dense in $\mathcal{C}_{\infty}(\bbR)$?
2. Let $\mathcal{C}_0([0,1])$ be the space of continuous functions on $[0,1]$ with supremum norm. Are there any continuous linear functionals $u:\mathcal{C}_0([0,1])\longrightarrow \bbC$ such that $u(fg)=u(f)u(g)$ for all $f,g\in\mathcal{C}_0([0,1]),$ where $fg(x)=f(x)g(x)$?
3. Let $L^1([0,1])$ and $L^2([0,1])$ be the Lebesgue spaces on $[0,1].$ What exactly is an element of each these spaces? What are their standard norms?
4. Which of $L^1([0,1])$ and $L^2([0,1])$ is a Hilbert space?
5. What are all the continuous linear functionals $u:L^1([0,1])\longrightarrow \bbC$ such that $u(fg)=u(f)u(g)$ for all $f,g\in L^2([0,1]).$
6. Let $u:\{(x,y);x^2+y^2<1\}\longrightarrow \bbC$ be a once differentiable function on the open unit ball which satisfies

   $$\frac{\pa u}{\pa x}+i\frac{\pa u}{\pa y}=0\Min x^2+y^2<1.$$

   
   
   Why is it true that $u$ is infinitely differentiable?
7. What functions are there as in the previous question which satisfy in addition $\frac{\pa ^ku}{\pa x^k}(0,0)=0$ for all $k$?
8. Every twice differentiable solution of the wave equation in two variables, $\frac{\pa^2u}{\pa t^2}-\frac{\pa^2u}{\pa x^2}=0$ in $\bbR^2$ is of the form $u_1(x+t)+u_2(x-t)$ for two twice differentiable functions of one variable.
9. There is no smooth map $f:\bbR\longrightarrow \bbR^2$ which is surjective.
10. For any sequence of real numbers $a_j,$ $j=0,1,\dots,$ there is a smooth function $u:\bbR\longrightarrow \bbR$ such that $\frac{d^j u}{dx^j}(0)=a_j$ for all $j.$