# Orthogonal complement

In the mathematical fields of linear algebra and functional analysis, the **orthogonal complement** of a subspace *W* of a vector space *V* equipped with a bilinear form *B* is the set *W*^{⊥} of all vectors in *V* that are orthogonal to every vector in *W*. Informally, it is called the **perp**, short for **perpendicular complement**. It is a subspace of *V*.

The definition extends to a bilinear form on a free module over a commutative ring, and to a sesquilinear form extended to include any free module over a commutative ring with conjugation.^{[1]}

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.

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.

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.