How do we define $df(a)$ for $a$ belong to the domain of $f$? A defination from Spivak Calculus On Manifold:
A function $f\colon \mathbb R^n \to  \mathbb R^m$ is differentiable at $a \in \mathbb R^n$ if there is a linear transformation $\lambda \colon \mathbb R^n \to \mathbb R^m$ such that $\lim_{h \to 0}\frac{|f(a+h)-f(a)-\lambda(h)|}{|h|} =0$.
They said the linear transformation $\lambda$ is denoted $Df(a)$ and called derivative of $f$ at $a$. So we can say $Df(a) \colon \mathbb R^n \to \mathbb R^m$
So my question how do we define $df(a)$? what is the domain and codomain? 

Given an $f$ and a point $a$ in the domain of $f$ there is at most one linear transformation $\lambda$ such that
$$f(a+h)-f(a)=\lambda.h +o\bigl(|h|\bigr)\qquad(h\to0)\ ,\tag{1}$$
because two different such transformations would differ by more than $o\bigl(|h|\bigr)$ when $h\to0$. It follows that the $\lambda$, if it exists, is uniquely determined by $f$ and $a$. It therefore can be denoted by $df(a)$ (this is just a typographical picture, not something infinitely small), or $Df(a)$, or similarly.
