Definitions

# Orthogonal complement

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

$W^bot=left\left\{xin V : langle x, y rangle = 0 mbox\left\{ for all \right\} yin W right\right\}.,$

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.,

$W^\left\{bot,bot\right\}=overline\left\{W\right\}.$

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:

$W^\left\{bot,bot\right\}=W.$

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

begin\left\{align\right\}
(mbox{Row},A)^bot &= mbox{Null},A (mbox{Col},A)^bot &= mbox{Null},A^T. end{align}

## 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

$W^bot = left\left\{,xin V^* : forall yin W x\left(y\right) = 0 , right\right\}.,$

It is always a closed subspace of $V^*$. There is also an analog of the double complement property. $W^\left\{bot,bot\right\}$ is now a subspace of $\left\{V^*\right\}^*$ (which is not identical to $V$). However, the reflexive spaces have a natural isomorphism $i$ between $V$ and $\left\{\left\{V^*\right\}^*\right\}$. In this case we have

$ioverline\left\{W\right\} = W^\left\{bot,bot\right\}.$

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

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