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Let $\begin{align*} X&=\mathcal{C}[0,1],\\ V&=\{v\in \mathcal{C}[0,1]\mid v(x)=v(-x)\},\\ W&=\{w\in\mathcal{C}[0,1]\mid w(x)=-w(-x)\}. \end{align*}$ Is it possible to verify that $X$ is a direct sum of $V$ and $W$? Someone help me...

Definition: $X$ is said to be the direct sum of $V$ and $W$ if $u\in X$ has a unique representation $u=v+w$ where $v\in V$ and $w\in W$.

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    Well, the definitions are certainly nonsensical...2012-06-05

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Everything would make sense if we consider $[-1,1]$ instead of $[0,1]$. The result means that we can write in a unique way a (continuous) function as a sum of an even and a odd continuous functions. To see that, write $u(x)=\underbrace{\frac{u(x)+u(-x)}2}_{=: v(x)}+\underbrace{\frac{u(x)-u(-x)}2}_{=:w(x)}.$

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    One down, [20.780 to go](http://math.stackexchange.com/unanswered) :-/2013-04-19