Dimension of $\ell^{1}$.

Question

I'm trying to figure out the dimension (cardinality of a Hamel Base) of the space $$\ell^1=\ell^{1}(\mathbb{N})=\left\lbrace x\colon \mathbb{N} \to \mathbb{R}\ \bigg\vert \ \sum_{i \in \mathbb{N}}|x(i)|<\infty \right\rbrace$$ and the only thing that I'know is that it must be greater or equal than $c = |\mathbb{R}|$ because is a infinite dimensional Banach space. A more general question would be ask for the dimension of $\ell^{1}(I)$ for a general set $I$, or even more general the dimension of $L^{1}(X,\mathcal{B},\mu)$.

Any help or reference will be appreciated.

Answer 1

Hint: What is the cardinality of $\ell^1$?