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 $w\in W$.