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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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    $v(x)=v(-x)$ for functions defined on $[0,1]$ ?2012-06-05
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    Your definitions make no sense: if $x\in [0,1]$, then $-x\in [0,1]$ if and only if $x=0$. Did you mean $\mathcal{C}[-1,1]$? If so, you are asking whether every continuous function can be decomposed (uniquely) as a sum of an even function and an odd function.2012-06-05
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    I think the question is wrong set as I doubt, I dont know how to fix it..2012-06-05
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    Well, the definitions are certainly nonsensical...2012-06-05

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