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Orthogonal complement

The subspace of vectors orthogonal to every vector in a given subset.

For a subset SS of an HH, its orthogonal complement is

S⊥={x∈H:⟨s,x⟩=0 for every s∈S}.S^\perp=\{x\in H:\langle s,x\rangle=0\text{ for every }s\in S\}.

It is always a . If MM is closed in a , then every x∈Hx\in H has a unique decomposition x=m+nx=m+n with m∈Mm\in M and n∈M⊥n\in M^\perp, so H=M⊕M⊥H=M\oplus M^\perp.