Taylor series operator is linear

This article gives a statement of the form that a certain operator from a space of functions to another space of functions is a linear operator, i.e., applying the operator to the sum of two functions gives the sum of the applications to each function, and applying it to a scalar multiple of a function gives the same scalar multiple of its application to the function.

Statement

Additivity and scalar multiples

1. Additivity: Suppose $f$ and $g$ are functions defined on subsets of reals that are both infinitely differentiable functions at a point $x_{0}$ which is in the domain of both functions. Then, $f+g$ is infinitely differentiable at $x_{0}$ and the Taylor series of $f+g$ about $x_{0}$ is the sum of the Taylor series of $f$ at $x_{0}$ and the Taylor series of math>g</math> at $x_{0}$.
2. Scalar multiples: Suppose $f$ is a function defined on a subset of the reals and it is infinitely differentiable at a point $x_{0}$ in its domain. Suppose $\lambda$ is a real number. Then, $\lambda f$ is infinitely differentiable at $x_{0}$ and the Taylor series of the function $\lambda f$ at $x_{0}$ is $\lambda$ times the Taylor series of $f$ at $x_{0}$.