What's the meaning of denoting $\ell^p$ as $\ell^p(\mathbb{N})$?

Question

What's the meaning of denoting $\ell^p$ as $\ell^p(\mathbb{N})$?

I read that it's like $\ell^p$ "over $\mathbb{N}$". But $l^p$ is sequences indexed by $\mathbb{N}$.

So it seems weird to treat the index set as an input to $\ell^p$? Since if $x_n$ is a function, then it could have much more as input or domain than $\mathbb{N}$.

Answer 1

You can define $\ell ^p$ to be a space of sequences with any set of indices (usually it is $\mathbb{N}$), so the space is depended on the set of indices.

Answer 2

Because, given a set $X$, $\ell^p(X)=L^p(X,\mathcal{P}(X),\Omega)$, where $\Omega$ is the counting measure (that is, $\Omega(A)=\#A$).