Orthogonal complement

In the mathematical fields of linear algebra and functional analysis, the orthogonal complement W^⊥ 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

Informally, it is called the perp, short for perpendicular 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. In such spaces, the orthogonal complement of the orthogonal complement of W is the closure of W, i.e.,

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 n, the orthogonal complement of a k-dimensional subspace is an (n − k)-dimensional subspace, and the double orthogonal complement is the original subspace:

If A is an m × n matrix, where Row A, Col A, and Null A refer to the row space, column space, and null space of A (respectively), we have

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 by

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.