orthogonal complement

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}.\,

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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