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Let $\mathcal{U}(4)$ be a subspace of $\mathcal{P}(4)$ consisting of polynomials that are even functions. Show that there exists a subspace $W \subset \mathcal{P}(4)$ such that $$\mathcal{P}(4) = \mathcal{U}(4) \oplus W.$$

How do I approach this problem? I'm not certain where I should start. I know that a function $f:\mathbb{R} \mapsto \mathbb{R}$ is even if $f(x) = f(-x)$ for all $x$.

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    $\begingroup$ This holds for any subspace of any vectorspace. $\endgroup$ Oct 12, 2013 at 19:32
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    $\begingroup$ Hint: Before you can answer a question, you have to understand it. $\endgroup$
    – Carsten S
    Oct 12, 2013 at 19:57
  • $\begingroup$ I just hate getting "hints" for answers if the user is not going to participate in a bit of discussion in the comments. This isn't homework. $\endgroup$
    – St Vincent
    Oct 12, 2013 at 20:05
  • $\begingroup$ Whether or not it is homework, it is clearly an exercise you have been given in some context (whether or not you will be graded on it or anyone will ever see your attempted solution is irrelevant). Hence, just getting a full solution will not help you learn as much as getting hints will. $\endgroup$ Oct 13, 2013 at 10:56
  • $\begingroup$ Sure, mindlessly reading a solutions manual would not be beneficial to an individual's learning. But what's so wrong with a complete solution to a problem I have already struggled with? Doesn't build enough character? I find that I learn plenty from complete solutions. Hints are fine too, but a "hint" should not be an excuse for a skimpy answer to gain quick reputation. $\endgroup$
    – St Vincent
    Oct 13, 2013 at 20:38

1 Answer 1

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Hint: $$\begin{align} f(x) = \frac{f(x)+f(-x)}{2} + \frac{f(x)-f(-x)}{2}. \end{align} $$

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  • $\begingroup$ Do I just assign $\begin{align} W = \frac{f(x)-f(-x)}{2} \text{ and } \mathcal{U}(4) = \frac{f(x)+f(-x)}{2} \end{align}$? $\endgroup$
    – St Vincent
    Oct 12, 2013 at 19:34
  • $\begingroup$ No. The function $\dfrac{f(x)-f(-x)}{2}$ is odd. You can write every polynomial as the sum of an even function and an odd function. $\endgroup$
    – njguliyev
    Oct 12, 2013 at 19:38
  • $\begingroup$ If you explain a bit further, I will give you the green check mark. Otherwise, I will leave this question unanswered. $\endgroup$
    – St Vincent
    Oct 13, 2013 at 2:55
  • $\begingroup$ Do you know what does $\mathcal{P}(4) = \mathcal{U}(4) \oplus W$ mean? It means that every polynomial can uniquely be written as the sum of two elements from $\mathcal{U}(4)$ and $W$, respectively. $\endgroup$
    – njguliyev
    Oct 13, 2013 at 7:53

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