Definition:P-Sequence Space/Real/Also denoted as

Real $p$-Sequence Space: Also denoted as
The real $p$-sequence space ${\ell^p}_\R$ is often denoted just as $\ell^p$ when there is no confusion as to what the underlying set is.

Some sources use the form $\map { {\ell^p}_\R} \N$ when it is necessary to bring attention to the fact that the domain of the sequences is the natural numbers.

That is, that the sequences in question are infinite.