Editing Orthogonal complement (section)

==Banach spaces==
There is a natural analog of this notion in general [[Banach space]]s. In this case one defines the orthogonal complement of <math>W</math> to be a subspace of the [[Dual space|dual]] of <math>V</math> defined similarly as the [[Dual space#Annihilators|annihilator]]
<math display="block">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^{\perp \perp}</math> is now a subspace of <math>V^{**}</math>(which is not identical to <math>V</math>). However, the [[reflexive space]]s have a [[natural transformation|natural]] [[isomorphism]] <math>i</math> between <math>V</math> and <math>V^{**}</math>. In this case we have <math display="block">i\overline{W} = W^{\perp\perp}.</math>

This is a rather straightforward consequence of the [[Hahn–Banach theorem]].