Soft question: what is the intuition behind labeling of metrics?

Question

I was wondering why taxicab, euclidean and sup norm metrics are labeled as $l^1$, $l^2$ and $l^{\infty}$ retrospectively? Are there any $l^3, l^4, \dots$-metrics defined in between?

Answer 1

Yes, for every real $p \ge 1$ there is a metric $d_p$ on a set of real or complex sequences:

$$d_p((x_n),(y_n)) = \sqrt[p]{\sum_{n=1}^\infty |x_n - y_n|^p}$$

which is defined on the set of sequences $(x_n)$ such that

$$\|(x_n)\|_p := \sqrt[p]{\sum_{n=1}^\infty |x_n|^p} < \infty$$

which is called $\ell^p$, and $\|\cdot\|_p$ is a norm and the set $\ell^p$ becomes a Banach space.

As $$\lim_{p \to \infty} \|x_n\|_p \to \sup \{ |x_n|: n = 1,2,3\ldots \}$$

the latter supremum is defined to be $\|x\|_{\infty}$, and the corresponding set of (bounded) sequences is called $\ell^\infty$, and is also a Banach space.

Even for $0 < p < 1$ we can define $\ell_p$ as the set of all sequences such that $\sum_n |x_n|^p$ converges and define a metric

$$d_p = \sum_{n=1}^\infty |x_n - y_n|^p$$

on this metric vector space, but it's not locally convex so not normable for these $p$. The standard sequence spaces $\ell_p$ for $p \ge 1$ are classic and well-studied spaces. It turns out that the dual space of $\ell^p$ is $\ell^q$ where $\frac{1}{p} + \frac{1}{q} = 1$. The metrics $d_p$ are also used on the finite sequences, i.e. $\mathbb{R}^n$. On a finite dimensional space all these norms are equivalent, but they're not on the infinite sequences.