Standard Basis of Function Space $\mathbb{R}^\mathbb{R}$

Question

I have seen $\mathbb{R}^\mathbb{R}$, the set of functions from $\mathbb{R}$ to $\mathbb{R}$, described as a vector space (with the usual operations $(f+g)(x)≔f(x)+g(x)$ and $(a\cdot f)(x)≔a\cdot f(x)$). However, I have been unable to find what the standard basis for this space is. Intuitively, I would think that the set of all functions that can be defined as $f_r(x)≔\begin{cases}1&\text{if }x=r\\0&\text{if }x\ne r\end{cases}$ for some $r\in\mathbb{R}$ would be the standard basis, as it is orthonormal and looks similar to the standard bases of the $\mathbb{R}^n$ spaces; however, I could not find this definition anywhere and was wondering if this might not be the case for some reason.

Answer 1

You have a couple of misconcenptions. Firstly, there is no the basis, a vector space can have many different bases (plural of basis). Secondly, what do you mean by orthogonal? I don't think the set of all functions from $\mathbb{R}$ has a particularly natural inner product.

Also, the set of functions you describe are certainly linearly independent, they don't span all the functions. Fx. $f(x)=x$. I'm pretty sure you won't be able to find any nice basis of this space, as the very existance of a basis uses the axiom of choice, which means it might not even to explicitly describe a basis.