Given the $\mathbb{R}$-vectorspace $V =\mathbb{R}^\mathbb{N}$ of real valued series I was wondering if we can construct a Hamel-basis of $V$.
First of all I think that the dimension of $V$ is $ \mid\mathbb{R}\mid$. Since $\mid V \mid = \mid \mathbb{R}^\mathbb{N}\mid = (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0 \times \aleph_0} = 2^{\aleph_0} =\mid \mathbb{R}\mid$ we get $dim(V) \leq \mid V \mid =\mid\mathbb{R}\mid$. But I am not sure why the dimension can't be countable. I can't seem to find an uncountable number of $\mathbb{R}$-linearly independent series.
Moreover I am interested if such a basis can ever be constructible. I doubt this to be the case, but I would like to know why not.
