3

A function $f : \mathbb{R} \rightarrow \mathbb{R}$ is called even if $f(-x) = f(x)$ $ \forall x \in \mathbb{R} $

A function $f : \mathbb{R} \rightarrow \mathbb{R}$ is called odd if $f(-x) = -f(x)$ $ \forall x \in \mathbb{R} $

Let $U_{e}$ denote the set of real-valued even functions on $\mathbb{R}$ and let $U_{o}$ denote the set of real valued odd functions on $\mathbb{R}$.

Show that $\mathbb{R}^{\mathbb{R}} = U_{e} \oplus U_{o}$

This question from Q24 of "Linear Algebra Done Right" 3rd Edition - Chp 1.4 (Subspaces). The author of the book does minimal work with functions during the chapter. I have no idea how to approach the question.

Asaf Karagila
  • 393,674
Dom
  • 343

2 Answers2

6

Hint: for each $f$ consider the even function $f(x)+f(-x)$.

Martín-Blas Pérez Pinilla
  • 41,546
  • 4
  • 46
  • 89
  • I'm still very confused as to how to finish the question. The author spends roughly 100 words on functions in up until this point. I know that f(x) + g(x) = (f+g)(x), but that is all... – Dom Dec 11 '14 at 11:07
  • 1
    @Dom, $f(x)=f(x)+f(-x)-f(-x)=\cdots$ – Martín-Blas Pérez Pinilla Dec 11 '14 at 11:10
  • Do you know of a resource that I can use to learn about this topic? Every single proof on functions in the textbook has been left for the reader as an exercise and there is no answer key for any of the exercises... – Dom Dec 11 '14 at 11:22
  • @Dom, start from the Wikipedia: http://en.wikipedia.org/wiki/Function_%28mathematics%29. For linear algebra, see http://www.math.hawaii.edu/~lee/linear/index.html. – Martín-Blas Pérez Pinilla Dec 11 '14 at 16:16
0

Hint : $f(x)=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}$, so if $$g(x)=\frac{f(x)+f(-x)}{2}, h(x)=\frac{f(x)+f(-x)}{2}$$

$g(x) $ is even and $h(x)$ is odd

erfan soheil
  • 1,089