Let $\mathbb{R}^\mathbb{N}$ denote the vector space over $\mathbb{R}$ of sequences of real numbers, with multiplication and addition defined by component. It's well-known that though the subspace $\mathbb{R}^\infty$ of sequences with only a finite number of nonzero terms has a basis $\mathbf{e}_1 = (1, 0, 0, 0, \ldots), \mathbf{e}_2 = (0, 1, 0, 0, \ldots)$, this is not a basis of $\mathbb{R}^\mathbb{N}$ (expressing the constant sequence $(1, 1, 1, \ldots)$ would require an infinite sum $\mathbf{e}_1 + \mathbf{e}_2 + \mathbf{e}_3 + \cdots$, and infinite sums in generic vector spaces are undefined). It's also been proved that the statement that all vector spaces have a basis is equivalent to the axiom of choice.
I'm interested, though, in the specific space $\mathbb{R}^\mathbb{N}$. Has it been proved that a basis for this set requires the axiom of choice and cannot be described explicitly? This isn't a homework question or anything; I'm just curious.