Let’s just quickly verify the condition. We need to show that if and are subspaces of an inner-product space , then if and only if . Clearly the symmetry of the situation shows us that we only need to check one direction. So if , we know that , and also that . And thus we see that .
So what does this tell us? First of all, it gives us a closure operator — the double orthogonal complement. It also gives a sense of a “closed” subspace — we say that is closed if .
But didn’t we know that ? No, that only held for finite-dimensional vector spaces. This now holds for all vector spaces. So if we have an infinite-dimensional vector space its lattice of subspaces may not be orthocomplemented. But its lattice of closed subspaces will be! So if we want to use an infinite-dimensional vector space to build up some analogue of classical logic, we might be able to make it work after all.}