= Orthogonal complement for a sesquilinear form
{title2=$W^\perp=\{v:B(v,w)=0\text{ for all }w\in W\}$}
For a <Hermitian form> or a <Hermitian form over a quadratic finite field>, the displayed subspace is the orthogonal complement of $W$. If the form restricts nondegenerately to a finite-dimensional $W$, its invertible Gram matrix supplies, for each $v$, a unique $w_0\in W$ with $B(v-w_0,W)=0$. Consequently $V=W\oplus W^\perp$. When the form on $V$ is nondegenerate, its restriction to $W^\perp$ is too: a radical vector there is orthogonal to both summands and hence is zero.
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