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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 $$\mathcal{P}(4) = \mathcal{U}(4) \oplus W.$$

Additionally, I know that a function $f:\mathbb{R} \mapsto \mathbb{R}$ is even if $f(x) = f(-x)$ for all $x$.

And that a function can also be expressed as the sum of an odd an even function $$f(x) = \frac{f(x)+f(-x)}{2} + \frac{f(x)-f(-x)}{2}.$$

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$$\mathcal{U}(4)=Span(1,x^2, x^4), \mathcal{W}=Span(x,x^3)$$

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  • $\begingroup$ Thanks! :) Clarified everything. $\endgroup$
    – St Vincent
    Commented Oct 13, 2013 at 18:13
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    $\begingroup$ My pleasure, happy to help. $\endgroup$
    – vadim123
    Commented Oct 13, 2013 at 18:14

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