Vector subspace

Orthogonal complement

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

Proof of subspace

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

Properties

Let be an arbitrary subset. Then

1. is topologically closed
5. If for some , 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.

Let be a vector subspace. Then

1. (Internal direct sum).
2. .

Proof of 1–2

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

Other properties include

• The orthogonal complement of an invariant subspace under a unitary operator is invariant

See also

• Orthogonal complement polarity

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