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

The dimension of $\mathbb R^\mathbb R$ over $\mathbb R$ is $2^{\frak c}$. It is not even the size of the continuum. As Jeroen says, this space is not finitely generated. Not even as an algebra.
Even as an algebra it is not finitely generated. What does that mean? Algebra is a vector space which has a multiplication operator. In $\mathbb R[x]$, the vector space of polynomials, we can write any polynomial as a finite sum of scalars and $x^n$'s. This is an example for a vector space which is not finitely generated, but as an algebra it is finitely generated.
In introductory courses it is customary to deal with well understood spaces. In the early beginning it is even better to use only finitely generated spaces, which are even better understood.
The axiom of choice is an axiom which allows us to "control" infinitary processes. Assuming this axiom we can prove that every vector space has a basis, but we cannot necessarily construct such space.
