of with complement Properties The orthogonal complement is always closed in the metric topology. In finite-dimensional spaces, that is merely an instance of the fact that all subspaces of a vector space are closed. In infinite-dimensional Hilbert spaces, some subspaces are not closed, but all orthogonal complements are closed. If is a vector subspace of an inner product space the orthogonal complement of the orthogonal complement of is the closure of that is, Some other useful properties that always hold are the following. Let be a Hilbert space and let and be its linear subspaces. Then: ; if then ; ; ; if is a closed linear subspace of then ; if is a closed linear subspace of then the (inner) direct sum. The orthogonal complement generalizes to the annihilator, and gives a Galois connection on subsets of the inner product space, with associated closure operator the topological closure of the span. Finite dimensions For a finite-dimensional inner product space of dimension the orthogonal complement of a -dimensional subspace is an -dimensional subspace, and the double orthogonal complement is the original subspace: If is an matrix, where and refer to the row space, column space, and null space of (respectively), then Banach spaces There is a natural analog of this notion in general Banach spaces. In this case one defines the orthogonal complement of W to be a subspace of the dual of V defined similarly as the annihilator It is always a closed subspace of V∗. There is also an analog of the double complement property. W⊥⊥ is now a subspace of V∗∗ (which is not identical to V). However, the reflexive spaces have a natural isomorphism i between V and V∗∗. In this case we have This is a rather straightforward consequence of the Hahn–Banach theorem. Applications In special relativity the orthogonal complement is used to determine the simultaneous hyperplane at a point of a world line. The bilinear form η used in Minkowski space determines a pseudo-Euclidean space of events. The origin and all events on the light cone are self-orthogonal. When a time event and a space event evaluate to zero under the bilinear form, then they are hyperbolic-orthogonal. This terminology stems from the use of two conjugate hyperbolas in the pseudo-Euclidean plane: conjugate diameters of these hyperbolas are hyperbolic-orthogonal. See also Notes References Bibliography External links Orthogonal complement ; Minute 9.00 in the Youtube Video