# 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

• – Sean Eberhard Dec 1 '13 at 21:55

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.
• The $l^0$ space (at least, the way it was defined for us), is the sequence space of "eventually zero" sequences, which is usually equipped with the uniform norm. – Andrew D Dec 1 '13 at 21:59
• @AndrewD Ah, I see. I usually call it $c_0$. – Daniel Robert-Nicoud Dec 1 '13 at 22:01
• I get the feeling that there isn't a standard definition for it, seeing as the wikipedia page defines $c_0$ as the sequence space of sequences with limit 0, whereas the "eventually zero" definition means that every sequence is the space is finite with limit 0. – Andrew D Dec 1 '13 at 22:05