Definition of p-th Summable Sequences (the Sequence Spaces of l^p, with p=2)

Question

Can someone please give detailed definitions for the following Sequence Spaces:

1. $l^2(\mathbb{N})$
2. $l^2(\mathbb{Z})$
3. $l^2(\mathbb{Z^n})$

I would greatly appreciate it if you would not use the phrase "p-th summable sequence" in your definition. I see a lot of this going on and it does not help.

Thank you.

Answer 1

For any set $A$ and any real number $p\geq 0$, $\ell^p(A)$ is defined to be the set of functions $f:A\to\mathbb{C}$ such that $$ \sum_{x\in A}|f(x)|^{p}<\infty $$ Note that since $|f(x)|^p$ is a non-negative real number, this sum can be unambiguously defined by taking the supremum of $\sum_{x\in F}|f(x)|^p$ over all finite subsets $F$ of $A$.

If $A$ is a finite set, say with $n$ elements, and $p\geq 1$, then $\ell^p(A)$ is just $\mathbb{C}^n$ with the $p$-norm.

If $A=\mathbb{N}$, then $\ell^p(\mathbb{N})$ is the set of sequences $\{f(n)\}_{n=1}^{\infty}$ for which $$ \sum_{n=1}^{\infty}|f(n)|^p<\infty $$

There are also real $\ell^p$ spaces, where the functions in question must be real-valued, but in my experience if an $\ell^p$ space is referred to with no further qualifications then the functions are assumed to be complex-valued.