Orthogonal complement

Given an inner product space , the orthogonal complement of a subset is the vector subspace of vectors orthogonal to those #m/def/linalg

Proof of subspace

Clearly\Span. If and then for all , and thus . Therefore is a subspace.

A slightly more general concept is Dual annihilator.

Properties

Let be an arbitrary subset. Then

Proof of 1–6

Note that the orthogonal complement of a singleton can be expressed as a preïmage

Since is closed in , and the inner product is continuous, it follows is closed. Now for an arbitrary set ,

which is an intersection of closed sets and is therefore closed, proving ^S1.

Note if then , implying by ^IP3, proving ^S2.

Let and . Then for any , so , proving ^S3.

Let . Then by definition for any , so , proving ^S4.

Without loss of generality , for iff . Now

and

but since , it follows from ^IP3 that , proving ^S5.

Let and , so for some . Then

proving ^S6.

Proof of 1–2

^V1 follows directly from ^S2, and ^V2 follows directly from ^S4.

Other properties include