Probability
The Analysis of Data, volume 1
Metric Spaces: Exercises
$
\def\P{\mathsf{P}}
\def\R{\mathbb{R}}
\def\defeq{\stackrel{\tiny\text{def}}{=}}
\def\c{\,|\,}
\def\bb{\boldsymbol}
\def\diag{\operatorname{\sf diag}}
$
B.6. Exercises
- Verify the claims made in Example B.4.3 regarding the Euclidean norm and Euclidean distance in $\R^{\infty}$.
- Construct a subset of $\R^{\infty}$ on which the Euclidean distance generalization is a distance function.
- 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 $\R^{[0,1]}$?
- Show that the $L_1$ and $L_2$ distance functions result in the same collection of open sets in the Euclidean space $\R$.
- 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 $k\in\mathbb{N}$?
- 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?