ThmDex – An index of mathematical definitions, results, and conjectures.

Result R2929 on D3998: Odd euclidean real function

Sum of odd functions is odd

Formulation 0

Let $f_1, \dots, f_N : \mathbb{R}^n \to \mathbb{R}^m$ each be an D3998: Odd euclidean real function.
Let $\sum_{n = 1}^N f_n$ be the D4344: Pointwise function sum of $f_1, \dots, f_N$.

Then $\sum_{n = 1}^N f_n$ is an D3998: Odd euclidean real function.

Proofs

Proof 0

Let $f_1, \dots, f_N : \mathbb{R}^n \to \mathbb{R}^m$ each be an D3998: Odd euclidean real function.
Let $\sum_{n = 1}^N f_n$ be the D4344: Pointwise function sum of $f_1, \dots, f_N$.

If $x \in \mathbb{R}^n$, then \begin{equation} \begin{split} \Big( \sum_{n = 1}^N f_n \Big) (- x) & = \sum_{n = 1}^N f_n(- x) \\ & = \sum_{n = 1}^N - f_n(x) = - \sum_{n = 1}^N f_n(x) = - \Big( \sum_{n = 1}^N f_n \Big) (x) \end{split} \end{equation} $\square$