Orthogonal halves of $C[0,1]$ that cannot rebuild a function non-zero at $1/2$

Let $V=\{f\in C[0,1]:f(t)=0 \forall t\le1/2\}$ and $W=\{f\in C[0,1]:f(t)=0 \forall t>1/2\}$. If $C[0,1]$ is assigned the inner product $\langle f,g\rangle=\int_0^1f(t)\overline{g(t)}dt$, then: