## Probability

### The Analysis of Data, volume 1

Metric Spaces: Exercises

## B.6. Exercises

1. Verify the claims made in Example B.4.3 regarding the Euclidean norm and Euclidean distance in $\R^{\infty}$.
2. Construct a subset of $\R^{\infty}$ on which the Euclidean distance generalization is a distance function.
3. Consider the space $\R^{[0,1]}$ of functions from $[0,1]$ to $\R$ (see Definition A.3.4). Is $\langle f,g\rangle=\int_0^1 f(x)g(x)\,dt$ a valid inner product? Is the natural generalization of the Euclidean distance a valid distance function on $\R^{[0,1]}$?
4. Show that the $L_1$ and $L_2$ distance functions result in the same collection of open sets in the Euclidean space $\R$.
5. What is the condition for a sequence ${\bb x}^{(n)}, n\in\mathbb{N}$ to converge to $\bb x$ in $(\R^{\infty},\bar d)$. Is it identical to the condition that $x^{(n)}_k\to x_k$ for all $k\in\mathbb{N}$?
6. Generalize the distance function $\bar d$ in Example B.4.4 to the set $\R^{[0,1]}$. Is the resulting space $(\R^{[0,1]},\bar d)$ a metric space?