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

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

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.,

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

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.