- #1

- 1,307

- 108

[tex] \ell^{\infty}=\left\{\alpha : \sup\left\{ \left| \alpha_n \right| : n\in\mathbb{N}\right\}<\infty \right\}, [/tex] normed by [tex]\| \alpha\|_{\infty}=\sup\left\{ \left| \alpha_n \right| : n\in\mathbb{N}\right\}[/tex]

is not separable, that is that it does not possess a countable dense basis. I do not well understand what it means for a space to be separable : does it mean that any element (e.g. any sequence, vector, function, ...) of that space can be expressed as either a linear combination of the elements (say, functions) of some countable (basis?) set or a limit thereof?

Would someone please clearly explain this topic that I might more fully understand it, and, perhaps, the concept of dual spaces: specifically, why is [itex] \ell^{\infty}=(\ell^{1})^*[/itex]? where X* denotes the dual space of X and where [itex] \ell^{1}[/itex] is the Banach space of sequences of complex numbers defined by

[tex] \ell^{1}=\left\{\alpha : \sum_{k=1}^{\infty} \left| \alpha_k \right| <\infty \right\}, [/tex] normed by [tex]\| \alpha\|_{1}=\sum_{k=1}^{\infty} \left| \alpha_k \right|[/tex]

And, why, despite this relationship, is [itex] (\ell^{\infty})^*=(\ell^{1})^{**}\neq \ell^{1}[/itex]? I realize that I have asked alot, but I would rather that sufficient information be put forth that I could join-in the discussion.

Thanks,

--Ben