Why is $\left(\frac{\partial}{\partial x_i}\right)f = \left(\frac{\partial f}{\partial x_i}\right)$

Question

i'm currently reading An Introduction to Morse Theory by Yukio Matsumoto and on p.62 it says

A vector field itself is sort of a differential operator, since it assigns to each point a "tangent vector" which is a differential operation. Let us differentiate $f$ with respect to the gradient vector field $X_f$:

$$X_f \cdot f = \left(\sum_{i=1}^m \frac{\partial f}{\partial x_i} \frac{\partial}{\partial x_i}\right)\cdot f = \sum_{i=1}^m \left( \frac{\partial f}{\partial x_i}\right)^2 \ge 0$$

I would love to understand why $$ \left(\frac{\partial}{\partial x_i}\right)\cdot f = \left( \frac{\partial f}{\partial x_i}\right)$$

Could anyone help me on this? Thank you very much.

Answer 1

It is just notation. See here for a short description.