Orthogonal complement
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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>
[edit] 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
- <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 <math>V</math>). However, the reflexive spaces have a natural isomorphism <math>i</math> between <math>V</math> 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.de:Komplement (lineare Algebra)
