In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of an inner product space V is the set of all vectors in V that are orthogonal to every vector in W, i.e. it is

In infinite-dimensional Hilbert spaces, it is of some interest to observe that every orthogonal complement is closed in the metric topology—a statement that is vacuously true in the finite-dimensional case. The orthogonal complement of the orthogonal complement of W is the closure of W, i.e.,

Banach spaces

There is a natural analog of this notion in general Banach spaces. In this case one defines the orthogonal complemet of W to be a subspace of the dual of V defined similarly by

It is always a closed subspace of V ^* . There is also an analog of the double complement property. is now a subspace of which is not identical to V. However, the reflexive spaces have a natural isomorphism i between V and . In this case we have

This is a rather straightforward consequence of the Hahn-Banach theorem.