Direct sums producing the vector space R->R

Question

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?

Answer 1

No, it is not a counter example, since$$e^x=\overbrace{\frac{e^x+e^{-x}}2}^{\text{even function}}+\overbrace{\frac{e^x-e^{-x}}2}^{\text{odd function}}.$$

Answer 2

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?