Q:

Definition of $\ell^\infty$

I have come across the space of bounded sequences denoted as $\ell^\infty$ in my course, but not a clear, concise definition. I have seen sometimes when these includes sequences in $\mathbb{R}$ that are bounded and sometimes when the sequence is in $\mathbb{R}^\infty$! So I presume this implies $\ell^\infty$ for is bounded sequences in $\mathbb{R}^n$ (i guess this can be generalised in metric spaces as well). Also it will be great if someone could mention a clear definition of $\ell^0$ and $\ell^1$ as my lecturer didnt do so.
Thanks 

A:

The space $\ell^\infty$ is the Banach space given by
$$\ell^\infty = \{f\in\mathbb{R}^\mathbb{N}|\ \|f\|_\infty<\infty\}$$
where
$$\|f\|_\infty = \sup_n |f_n|$$
The definition for $\ell^p$, $1\leq p<\infty$ is similar, only using the norm
$$\|f\|_p=\left(\sum_{n=0}^\infty|f_n|^p\right)^{\frac{1}{p}}$$
I don't know of any $\ell^0$ space.

