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

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 $x_{0}$ .
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 $λ$ is a real number. Then, $λ f$ is infinitely differentiable at $x_{0}$ and the Taylor series of the function $λ f$ at $x_{0}$ is $λ$ times the Taylor series of $f$ at $x_{0}$ .