Is it true that $c_0$ is isomorphic to $l^1$ as a vector space?

Question

Let $c_0$ be the vector space of complex sequences converging to $0,$ taken with the sup-norm.

Is it true that $c_0$ is isomorphic to $l^1$ as a vector space?

Update: Is it also true that $c_0$ is isomorphic to its dual space as a vector space?

$\ell_1$ is the dual of $c_0$, so I don't understand your update. — Tomasz Kania, Jul 02 '16 at 10:21
@ErginSuer, no infinite-dimensional vector space is vector-space isomorphic to its algebraic dual... — Tomasz Kania, Jul 02 '16 at 10:26
Is my answer satisfactory? If not, please let me know and I will elaborate more. — Tomasz Kania, Aug 08 '16 at 20:46

score 2 · Answer 1 · edited Apr 13 '17 at 12:20

Yes, because they have cardinality continuum and, being infinite-dimensional Banach spaces, they have Hamel bases of cardinality at least continuum hence of cardinality exactly continuum. Two vector spaces over the same field are isomorphic if and only if they have bases of the same cardinality.

Arguing in the same way, we conclude that even $c_0$ and $\ell_\infty$ are isomorphic as vector spaces.

Is it true that $c_0$ is isomorphic to $l^1$ as a vector space?

1 Answers1