Is $l^2$ is separable Banach space?
I had proved the following 2 lemmas:
1) $l^p$ for $1\leq p<\infty $ is complete
2)$l^p$ for $1\leq p<\infty $ is separable.
With this, I can conclude that $l^2$ is separable Banach space.
But I also proved using Baire Category theorem that any infinite dimensional Banach space must have Hamel basis uncountable.
But as space is separable it must have a countable basis.
Where I am missing?
Please help me.
Best Answer
$\ell^2$ is a separable Banach space indeed (even Hilbert space, because it has inner product to give the norm).
It has a countable topological base and a countable orthonormal base (which is also a Schauder base, which is its generalisation in Banach spaces) but an uncountable linear vector space basis (aka Hamel basis). These are different notions of bases so there is no contradiction.