Please, help find out the truth or falsity of this:

Let $\displaystyle (X_i)_{i\in J}$ -family of subsets in V,V - an inner product space. $\displaystyle X_i\perp X_j, \quad X_i+X_i^\perp=V \quad \forall i$, and let $\displaystyle X=\overline{sp\bigcup X_i}$. Can we say that $\displaystyle X+X^\perp=V$?