Does $\mathbb{R}^\mathbb{R}$ have a basis? I'm studying linear algebra on my own and saw basis for $\mathbb{R}, \mathbb{R}[X], ...$ but there is no example of $\mathbb{R}^\mathbb{R}$ (even though it is used for many examples). What is a basis for it? Thank you

For this space (or simpler the vector space of infinite sequences of reals) one cannot actually write down a basis, even though one can prove using the axiom of choice that every vector space has a basis. Proofs of existence that essentially need the axiom of choice provide no means to actually construct the objects whose existence they establish. 
