A few days ago we were briefly discussing Taylor's theorem in higher dimensions in the lecture. Referring to the expression
$f(x)=f(a)+Df(a)(x-a)+$higher order
the lecturer said that in general $Df$ is a linear functional, but it becomes a map if $f$ is defined on a Banach space.
Now that I look at my notes, I don't really understand the statement. Is there a difference between a linear functional and a map? (in particular, why is it of importance here?)
Thank you
A linear functional is usually an element of the dual vector space, e.g. if you're working with real vector spaces, it's a map $f : V \to \mathbb R$. A map is just a function between sets.
EDIT : It might be that your author thinks of functionals as continuous linear maps, and in the case of Banach spaces it is not always the case that linear maps are continuous. Either way, I would need to read that part of the book to be more precise.
Hope that helps,