Orthogonal complement - Definition and Overview

In the mathematical fields of linear algebra and functional analysis, the orthogonal complement <math>W^\bot<math> 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

<math>W^\bot=\left\{\,x\in V : \forall y\in W\ \langle x \mid y \rangle = 0 \, \right\}.\, <math>

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.,

<math>W^{\bot\bot}=\overline{W}.\, <math>

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

<math>W^\bot = \left\{\,x\in V^* : \forall y\in W\ x(y) = 0 \, \right\}.\, <math>

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

<math>i\overline{W}=W^{\bot^\bot}.<math>

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