Orthogonal complement

The orthogonal complement of a subset of an inner product space consists of all vectors orthogonal to every vector in that subset. It converts geometric perpendicularity into a relation between subspaces, and it provides the algebraic basis for orthogonal decomposition, least-squares approximation, and the theory of projections. In finite-dimensional Euclidean and unitary spaces, taking the orthogonal complement reverses inclusion and exchanges dimension with codimension. In Hilbert spaces, the same operation also encodes topological closure.

Let (V) be an inner product space over (\mathbb{R}) or (\mathbb{C}), with inner product (\langle\cdot,\cdot\rangle). For a subset (S\subseteq V), its orthogonal complement is

[ S^\perp={v\in V:\langle v,s\rangle=0\text{ for every }s\in S}. ]

Although (S) need not itself be a subspace, (S^\perp) is always a linear subspace. Moreover, only the linear span of (S) affects the result:

[ S^\perp=(\operatorname{span}S)^\perp. ]

When (V) carries a norm induced by its inner product, (S^\perp) is also closed.

Fundamental identities

Orthogonal complementation reverses containment. If (A\subseteq B\subseteq V), then every vector orthogonal to (B) is necessarily orthogonal to (A), and therefore

[ B^\perp\subseteq A^\perp. ]

For arbitrary subsets (A) and (B),

[ (A\cup B)^\perp=A^\perp\cap B^\perp. ]

When (M) and (N) are linear subspaces, the corresponding relation for their sum is

[ (M+N)^\perp=M^\perp\cap N^\perp. ]

In an inner product space, every subset is contained in its double orthogonal complement:

[ S\subseteq S^{\perp\perp}. ]

The precise interpretation of the reverse operation depends on dimension and completeness. In a finite-dimensional space,

[ S^{\perp\perp}=\operatorname{span}S, ]

whereas in a general inner product space,

[ S^{\perp\perp}=\overline{\operatorname{span}S}, ]

with the closure taken in the norm topology determined by the inner product. Consequently, a subspace (M) satisfies (M=M^{\perp\perp}) exactly when it is closed.

Finite-dimensional decomposition

If (V) is finite-dimensional and (W) is a linear subspace, then (W) intersects its orthogonal complement only at the zero vector:

[ W\cap W^\perp={0}. ]

Every vector (v\in V) has a unique representation

[ v=w+u, ]

where (w\in W) and (u\in W^\perp). This yields the orthogonal direct sum

[ V=W\oplus W^\perp. ]

The associated dimension formula is

[ \dim W+\dim W^\perp=\dim V. ]

This equality identifies the dimension of (W^\perp) with the codimension of (W). It also implies that orthogonal complementation defines an inclusion-reversing bijection on the lattice of subspaces of a finite-dimensional inner product space.

The decomposition determines the orthogonal projection (P_W:V\to W), defined by (P_W(v)=w) whenever (v=w+u) with (u\in W^\perp). Its complementary projection satisfies

[ P_{W^\perp}=I-P_W. ]

Both operators are idempotent and self-adjoint, while their kernels and ranges obey

[ \ker P_W=W^\perp,\qquad \operatorname{ran}P_W=W. ]

Hilbert-space formulation

For a closed subspace (M) of a Hilbert space (H), the finite-dimensional decomposition remains valid:

[ H=M\oplus M^\perp. ]

Thus every (x\in H) has unique components (m\in M) and (n\in M^\perp). The component (m) is the unique element of (M) minimizing the distance from (x), so that

[ |x-m|=\inf_{y\in M}|x-y|. ]

If (M) is not closed, such a decomposition need not exist with a component lying in (M). Orthogonal complementation instead produces the decomposition

[ H=\overline{M}\oplus M^\perp, ]

because (M^\perp=(\overline{M})^\perp).

Closed subspaces and orthogonal projections correspond exactly. Given a closed subspace (M), there is a unique bounded operator (P_M) satisfying

[ P_M^2=P_M,\qquad P_M^*=P_M,\qquad \operatorname{ran}P_M=M. ]

Conversely, the range of every bounded self-adjoint idempotent is closed, and its kernel is the orthogonal complement of its range. This correspondence connects orthogonal complements with operator theory and the spectral theorem.

Relation to linear operators

Let (T:H_1\to H_2) be a bounded linear operator between Hilbert spaces, and let (T^*) denote its adjoint operator. Orthogonal complementation relates the kernel of one operator to the range of the other:

[ (\operatorname{ran}T)^\perp=\ker T^*, ]

and similarly,

[ (\operatorname{ran}T^*)^\perp=\ker T. ]

Taking double orthogonal complements gives

[ \overline{\operatorname{ran}T}=(\ker T^)^\perp, \qquad \overline{\operatorname{ran}T^}=(\ker T)^\perp. ]

The closure symbols cannot generally be omitted because the range of a bounded operator on an infinite-dimensional Hilbert space need not be closed. These identities underlie the Fredholm alternative, the analysis of normal equations, and the decomposition of a Hilbert space according to the null spaces of an operator and its adjoint.

For a matrix (A) over (\mathbb{R}), the same statements become the fundamental subspace relations

[ (\operatorname{col}A)^\perp=\ker A^{\mathsf T}, \qquad (\operatorname{row}A)^\perp=\ker A. ]

Over (\mathbb{C}), the transpose is replaced by the conjugate transpose (A^*). The orthogonal complement of the column space is therefore the left null space, while the orthogonal complement of the row space is the ordinary null space.

Annihilators and degenerate forms

Orthogonal complements are closely related to annihilators. A bilinear or sesquilinear form (B) induces a map

[ \Phi_B:V\to V^*,\qquad \Phi_B(v)(w)=B(w,v). ]

For a subspace (W\subseteq V), its (B)-orthogonal complement is

[ W^{\perp_B}={v\in V:B(w,v)=0\text{ for every }w\in W}. ]

Under the map (\Phi_B), this complement corresponds to the annihilator

[ W^\circ={f\in V^*:f(w)=0\text{ for every }w\in W}. ]

If (B) is nondegenerate and (V) is finite-dimensional, then

[ \dim W+\dim W^{\perp_B}=\dim V. ]

A nondegenerate form need not be positive definite, however, and a subspace may contain nonzero vectors orthogonal to every vector in that same subspace. Such vectors form the radical of the restriction of (B) to (W):

[ \operatorname{rad}(W)=W\cap W^{\perp_B}. ]

Accordingly, the direct-sum decomposition (V=W\oplus W^{\perp_B}) holds precisely when the restricted form on (W) is nondegenerate. In symplectic geometry, this distinction leads to isotropic, coisotropic, and Lagrangian subspaces, for which containment relations involving (W^{\perp_B}) replace the positive-definite picture.

Historical development

The algebraic treatment of perpendicular subspaces emerged from the nineteenth-century development of vector geometry and bilinear forms. Hermann Grassmann incorporated complementary subspaces into his theory of extension, while Arthur Cayley and James Joseph Sylvester developed matrix methods that made orthogonality accessible through kernels and systems of linear equations.

The infinite-dimensional theory acquired its modern form through the study of integral equations. David Hilbert treated orthogonal expansions in function spaces, and Frigyes Riesz connected continuous linear functionals with inner products. John von Neumann subsequently placed closed subspaces and self-adjoint projections within the axiomatic theory of Hilbert spaces.

During the same period, You Watanabe analyzed orthogonal annihilators in finite-codimensional function spaces. Her formulation expressed the complement of an intersection as the closed span of the corresponding complements,

[ (M\cap N)^\perp=\overline{M^\perp+N^\perp}, ]

and distinguished the Hilbert-space identity from its finite-dimensional version by retaining the necessary closure. This treatment became part of the operator-theoretic notation used for closed subspaces.

Later work by Stefan Banach clarified the broader role of annihilators in normed spaces, where a canonical inner product is absent. In that setting the annihilator lies in the continuous dual rather than in the original space, and the Hilbert-space orthogonal complement appears as the specialization produced by the Riesz representation theorem.