Ir al contenido principal

## Due August 14

### Problem 1

The function $d_T(x,y) = |x^1 - y^1| + \ldots + |x^n - y^n|$ defined for $x,y\in\R^n$ is a metric on $\R^n$.

### Problem 2

Two norms $||\cdot||_1$ and $||\cdot||_2$ on a vector space are equivalent if there exist constants $c_1, c_2$ such that

$c_1||x||_1 \le ||x||_2 \le c_2||x||_1$

for all $x\in X$.

1. The norms $||\cdot||_E, ||\cdot||_M$ and $||\cdot||_T$ on $\R^n$ are equivalent.

2. If $||\cdot||_1$ and $||\cdot||_2$ are equivalent and $B^i_r(x)$ is the ball of radius $r$ with center $x$ with respect to the metric induced by $||\cdot||_i$, then, for each $\e>0$, there exist $\delta_1,\delta_2>0$ such that $B^1_{\delta_1}(x) \subset B_\e^2(x)$ and $B^2_{\delta_2}(x)\subset B_\e^1(x)$.

3. Let $||\cdot||_1$ and $||\cdot||_2$ two norms on $X$, and suppose there exist $\delta, \e>0$ such that $B^1_\delta(0) \subset B_1^2(0)$ and $B_\e^2(0)\subset B_1^1(0)$. Then $||\cdot||_1$ and $||\cdot||_2$ are equivalent.

4. Equivalent norms induce the same topology; i. e., $U$ is open in $(X,||\cdot||_1)$ if and only if it is open in $(X,||\cdot||_2)$.

### Problem 3

Let $||\cdot||_1$ and $||\cdot||_2$ be norms on $\R^n$ induced by inner products. Then they are equivalent.

### Problem 4

The uniform, $L^1$ and $L^2$ norms on $C([0,1])$ satisfy

$||f||_1 \le ||f||_2 \le ||f||_u$

for all $f\in C([0,1])$. However, not two of them are equivalent.

### Problem 5

Let $\bar B_\e(x)$ be the closed ball with center $x$ and radius $\e>0$, and $\overline{B_\e(x)}$ the closure of the open ball $B_\e(x)$. Then $\overline{B_\e(x)} \subset \bar B_\e(x)$. Give an example where $\bar B_\e(x) \not\subset \overline{B_\e(x)}$.