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

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

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.,

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

[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