Orthogonal complement

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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\{\,x\in V : \forall y\in W\ \langle x \mid y \rangle = 0 \, \right\}.\,

The orthogonal complement is always closed in the metric topology. In Hilbert spaces, the orthogonal complement of the orthogonal complement of W is the closure of W, i.e.,

W^{\bot\,\bot}=\overline{W}.\,

[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

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

It is always a closed subspace of V * . There is also an analog of the double complement property. W^{\bot\,\bot} 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

i\overline{W} = W^{\bot\,\bot}.

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

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