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?

  • $\begingroup$ What is the vector space? $\endgroup$
    – user722227
    Jan 26 '20 at 16:10
  • $\begingroup$ 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 $V=U_o\cup U_e$, $\endgroup$
    – almagest
    Jan 26 '20 at 16:12
  • $\begingroup$ What is "$\mathbb{R} \to \mathbb{R}$? The space of arbitrary functions $f:\mathbb{R} \to \mathbb{R}$? Continuous ones? Linear ones (i.e., $\Hom(\mathbb{R}, \mathbb{R})$? $\endgroup$
    – anomaly
    Jan 26 '20 at 16:17
  • $\begingroup$ Set of Real valued functions mapping real numbers to real numbers $\endgroup$ Jan 26 '20 at 16:44

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

  • $\begingroup$ This confuses me, because you're using e^x to define e^x. Seems a bit circular. What also confused me is that the sum of an even functions an odd function is odd. But looking at the graph of e^x we can see it's neither even or odd. Also see here : quora.com/Is-e-x-an-odd-or-an-even-function $\endgroup$ Jan 26 '20 at 16:38
  • $\begingroup$ 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 $e^x$. $\endgroup$ Jan 26 '20 at 16:39
  • $\begingroup$ Apologies I did not mean to offend. I just mean that the graph of the function and other explanations point to e^x being not even or odd. $\endgroup$ Jan 26 '20 at 16:50
  • $\begingroup$ 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. $\endgroup$ Jan 26 '20 at 16:58
  • $\begingroup$ Ahah! Thanks alot man! I see where my reasoning was wrong now. $\endgroup$ Jan 26 '20 at 18:48

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?

  • $\begingroup$ This is a power series, so the cardinality of the number of expressions is infinite. So we can't say anything about the evenness or oddness of the function from that series I think. $\endgroup$ Jan 26 '20 at 16:46
  • $\begingroup$ You can, and even if you couldn’t you could find the functions (hyperbolic sine/cosine) that they are the Taylor series of. $\endgroup$
    – user722227
    Jan 26 '20 at 20:41

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