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 (Halmos 1974, p. 123):
Informally, it is called the perp, short for perpendicular complement.
Contents |
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.,
Some other useful properties that always hold are:
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.
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
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.