Orthogonal complement

In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W^⊥ of all vectors in V that are orthogonal to every vector in W. Informally, it is called the perp, short for perpendicular complement. It is a subspace of V.

Let ${\displaystyle V}$ be a vector space over a field ${\displaystyle F}$ equipped with a bilinear form ${\displaystyle B}$. We define ${\displaystyle u}$ to be left-orthogonal to ${\displaystyle v}$, and ${\displaystyle v}$ to be right-orthogonal to ${\displaystyle u}$, when ${\displaystyle B(u,v)=0}$. For a subset ${\displaystyle W}$ of ${\displaystyle V}$ we define the left orthogonal complement ${\displaystyle W^{\bot }}$ to be

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