I am starting to read a book in Morse theory and Floer Homology. In the first few pages of morse theory I found a definition of second order derivative for functions $f \colon M \to \mathbb{R}$.
However, I am very confused with it. Here it is:
IN this case, he defines $X$ and $Y$ to be tangent vectors. BUt, as far as I know a tangent vector is such that it maps elements in $C^\infty(V)$ (here $V$ is a manifold) to $\mathbb{R}$. So here the $Y \circ f$ would be a number and does not make sense to me that it is being evaluated in $X$. Or in this case we should take them as vector fields? I am confused and I hope you can help me here.