What is the difference of p-norm between on $\mathbb{F}^n$ and on $\ell^p$

Question

The note from U. of Washington defines two $\ell^p$-norms on two different spaces $\mathbb{F}^n$ and $\ell^p$ (where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$). As far as I know, $\ell^p$ is a subspace of $\mathbb{F}^\infty$ (written on the second page), but what is the main difference between these two spaces for $p$-norm ? Any simple explanation? I can't tell the difference.

Answer 1

Assuming @xavierm02's interpretation of the symbols in the notes are correct, then you are right that the definition is the same for the two spaces.

The only difference is that $\ell^p$ is defined to be the subspace of $\mathbb{F}^{\infty}$ for which the $p-$norm is always defined, whereas the $p-$norm may not be defined for some (most) of the sequences in $\mathbb{F}^{\infty}$ in general. (see (2) on the second page of the notes)

$$|| (x_n) ||_p := \left[\sum_{i=0}^{\infty} |x^n|^p \right]^{\frac{1}{p}}$$

Note that $\mathbb{F}^n \not= \mathbb{F}^{\infty}$ for any finite $n$, so although the definition of a $p-$norm for $\mathbb{F}^n$ is exactly analogous to that for $\mathbb{F}^{\infty}$, there are some differences (1) the sum in the definition for $\mathbb{F}^n$ will always be finite, whereas it may be an infinite series for $\mathbb{F}^{\infty}$, and (2) the $p-$norm of $\mathbb{F}^n$ is defined on all of $\mathbb{F}^n$, whereas the $p-$norm of $\mathbb{F}^{\infty}$ is only defined on the subspace $\ell^{p} \subset \mathbb{F}^{\infty}$.

It's also worth noting that, in a strict sense, a norm is supposed to be a function on a space, thus needs to be defined everywhere. So unless you are mapping into the extended real (complex) numbers (i.e. with $\{\pm \infty\}$ or $\{\infty\}$, the point at infinity, added), then the $p-$norm can only be a function on $\mathbb{F}^n$ or $\ell^p$ but not $\mathbb{F}^{\infty}$.

Also, as @xavierm02 noted in the comments, there is for any $n$ is an injection of $\mathbb{F}^n$ into $\ell^p$ formed by mapping any vector in $\mathbb{F}^n$ to the sequence whose first $n$ entries are the entries of the vector and whose remaining (infinitely many) entries are all $0$.