# Homework 7, Real Analysis

## Due October 5

### Problem 1

Let $L^1([a,b])$ be the space of real valued continuous functions with the $d_1$ metric.

1. The polynomials are dense in $L^1([a,b])$.
2. Is $L^1([a,b])$ separable?

### Problem 2

Let $f:[a,b]\to\R$ be a continuous function such that

$\displaystyle \int_a^b f(x) x^n dx = 0$

for all $n=0,1,2,\ldots$. Then $f(x)=0$ for all $x\in[a,b].$

### Problem 3

If $X,Y$ are compact metric spaces, then the tensor space

$\displaystyle C(X)\otimes C(Y) = \{ (x,y)\mapsto \sum_{k=1}^n f_k(x)g_k(y): f_k\in C(X), g_k\in C(Y), n\ge1\}$

is dense in $C(X\times Y)$.

Note: The product space $X\times Y$ has the metric

$d_{X\times Y} \big( (x_1,y_1), (x_2,y_2) \big) = d_X(x_1,x_2) + d_Y(y_1,y_2).$

### Problem 4

State whether the following are true:

1. $\overline{A\cup B} \subset \overline{A}\cup \overline{B}$;
2. $\overline{A\cup B} \supset \overline{A}\cup \overline{B}$;
3. $\overline{A\cap B} \subset \overline{A}\cap \overline{B}$; and
4. $\overline{A\cap B} \supset \overline{A}\cap \overline{B}$.

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