# Direct sums producing the vector space R->R

An exercise in my assignment asks for the following proof.

Prove that the vector space R->R is equal to the direct sum of:

• the set of real valued even functions on R Ue

• the set of real valued odd functions on R Uo

Investigating this question I seem to have found a counter example f(x)= e^x.

This is because f(-x) = 1/e^x which is neither even or odd.

Can someone please clarify if I'm mistaken or if the exercise is wrong?

• What is the vector space? Jan 26 '20 at 16:10
• But $e^x=\cosh x+\sinh x$, and $\cosh x$ is even, $\sinh x$ is odd. $V=U_o\oplus U_e$ does not mean that $V=U_o\cup U_e$, Jan 26 '20 at 16:12
• What is "$\mathbb{R} \to \mathbb{R}$? The space of arbitrary functions $f:\mathbb{R} \to \mathbb{R}$? Continuous ones? Linear ones (i.e., $\Hom(\mathbb{R}, \mathbb{R})$? Jan 26 '20 at 16:17
• Set of Real valued functions mapping real numbers to real numbers Jan 26 '20 at 16:44

No, it is not a counter example, since$$e^x=\overbrace{\frac{e^x+e^{-x}}2}^{\text{even function}}+\overbrace{\frac{e^x-e^{-x}}2}^{\text{odd function}}.$$
• No, I am not. I am expressing $e^x$ as the sum of an even function with an odd function. At no point I defined the meaning of $e^x$. Jan 26 '20 at 16:39
• Of course it's neither even nor odd. And… ? Asserting that the space $\mathcal F$ of all functions from $\mathbb R$ into $\mathbb R$ is the direct sum of the space of all even functions with the space of all odd functions just means that every element of $\mathcal F$ can be written in one and only one way as the sum of an even function with an odd one. It does not mean that every element of $\mathcal F$ is either even or odd. Jan 26 '20 at 16:58
Consider the Taylor series for $$e^x$$: $$e^x=\sum_{i=0}^{\infty}\frac{x^n}{n!}$$. Then $$e^x$$ is the sum of the even and odd functions $$\sum_{i=0}^\infty\frac{x^{2n}}{n!}$$ and $$\sum_{i=1}^\infty\frac{x^{2n+1}}{n!}$$. Now what are those functions explicitly?