Please, help find out the truth or falsity of this:
Let (X_i)_{i\in J} -family of subsets in V,V - an inner product space. X_i\perp X_j, \quad X_i+X_i^\perp=V \quad \forall i, and let X=\overline{sp\bigcup X_i}. Can we say that X+X^\perp=V?