Orthogonal projection is the mechanism that turns perpendicularity into an actual operation. In the closed-subspace setting of a [Hilbert Space](/page/Hilbert%20Space), a subspace is not merely a subset: it determines a canonical nearest-point map, a decomposition of the ambient space, and a self-adjoint idempotent operator. In Euclidean geometry this is the familiar shadow of a vector on a line or plane. In analysis, the same idea supports [Fourier series](/page/Fourier%20Series), least-squares approximation, spectral theory, [conditional expectation](/page/Conditional%20Expectation) in $L^2$, and the fibre-metric geometry developed in [Fibre Bundles I: Bundles, Sections, and Transition Data](/page/Fibre%20Bundles%20I%3A%20Bundles%2C%20Sections%2C%20and%20Transition%20Data).

The word "projection" by itself only says that applying the map twice changes nothing. Orthogonality adds the crucial metric condition: the error vector is perpendicular to the chosen subspace. That extra condition is what makes the projection canonical. Without it, many different complementary subspaces produce many different projections onto the same range; with it, a closed subspace of a Hilbert space determines a unique [bounded linear operator](/page/Bounded%20Linear%20Operator).

The concept has two parallel forms. The Hilbert-space form treats orthogonal projection as an operator $P \in \mathcal{L}(H)$ satisfying $P^2=P$ and $P^*=P$, where $\mathcal{L}(H)$ denotes the space of bounded linear operators from $H$ to itself and $P^*$ denotes the adjoint operator of $P$. The geometric form treats it as the nearest-point map onto a closed subspace. In finite-dimensional linear algebra and smooth geometry, the same construction appears fibrewise: if a vector bundle $E \to M$ carries a smooth fibre metric and $F \subset E$ is a smooth subbundle, then each fibre splits into $F_p$ and its orthogonal complement, producing a smooth bundle map $P_F:E\to E$.

## Definition

The starting point is a Hilbert space, where inner products make it meaningful to ask for perpendicular errors. The key operation is not merely a decomposition theorem but a map: each vector should be assigned its component in the target subspace. This is a strong demand in infinite-dimensional spaces, and it is precisely why closed subspaces are the natural domains of the theory.

[definition: Orthogonal Projection onto a Closed Subspace] Let $H$ be a Hilbert space and let $M \subset H$ be a closed linear subspace. The orthogonal projection onto $M$ is the map $P_M:H\to H$ whose value at $x \in H$ is denoted by $P_Mx$, defined by the condition that $P_Mx \in M$ and \begin{align*} (x-P_Mx,m)_H=0 \end{align*} for every $m \in M$. [/definition]

The definition forces the error $x-P_Mx$ to have no component in any allowed direction $m\in M$. To describe those possible error vectors without repeatedly mentioning every test vector in $M$, one needs a separate object: the subspace of all vectors perpendicular to $M$. That object is the orthogonal complement, and it is the language in which projection decompositions are stated.

[definition: Orthogonal Complement] Let $H$ be a Hilbert space and let $M \subset H$ be a linear subspace. The orthogonal complement of $M$ is \begin{align*} M^\perp := \{x \in H : (x,m)_H = 0 \text{ for every } m \in M\}. \end{align*} [/definition]

The orthogonal complement is the candidate space for all possible projection errors. If every $x \in H$ can be written as a sum of a vector in $M$ and a vector in $M^\perp$, then the component in $M$ is forced to be the projected point. This is the content needed to justify the definition above: the Hilbert space [projection theorem](/theorems/1985) guarantees that the required vector $P_Mx$ exists and is unique. The error $x-P_Mx$ is not an arbitrary residual; it is orthogonal to every allowable correction in $M$.

Analysts often identify orthogonal projections by algebraic properties of the operator. This version is compact and powerful because it can be checked without first naming the range as a subspace.

[definition: Orthogonal Projection Operator] Let $H$ be a Hilbert space. A bounded linear map \begin{align*} P:H &\to H \end{align*} is an orthogonal projection operator if $P \in \mathcal{L}(H)$ and \begin{align*} P^2 &= P, & P^* &= P. \end{align*} [/definition]

The equation $P^2=P$ says that the operator is a projection; once a vector has been projected, projecting it again does not move it. The equation $P^*=P$ says that the projection is compatible with the [inner product](/page/Inner%20Product). Together they distinguish orthogonal projections from oblique projections.

In differential geometry and geometric analysis, orthogonal projection is used point by point on vector bundles. The definition is the same fibrewise idea, but smoothness matters because the projection must vary smoothly with the base point.

[definition: Orthogonal Projection onto a Subbundle] Let $E \to M$ be a smooth real vector bundle equipped with a smooth fibre metric $g$, and let $F \subset E$ be a smooth vector subbundle. The orthogonal projection onto $F$ is the smooth bundle map \begin{align*} P_F:E &\to E \end{align*} such that, for each $p \in M$, the restriction \begin{align*} (P_F)_p:E_p &\to E_p \end{align*} is the orthogonal projection of the [inner product space](/page/Inner%20Product%20Space) $(E_p,g_p)$ onto $F_p$. [/definition]

The fibrewise condition says that each vector $v \in E_p$ decomposes as a component in $F_p$ plus a component in the fibrewise orthogonal complement $F_p^\perp$. Smoothness is not a decorative condition: it is what allows $P_F$ to act on smooth sections and appear in differential operators.

## Equivalent Characterisations

The decomposition definition is geometric, while the operator definition is algebraic. In practice one often meets an endomorphism $P$ before knowing whether it comes from a chosen subspace and its orthogonal complement. The obstruction is that idempotence alone only says vectors split into a range and a kernel; it does not force those two pieces to be perpendicular. Self-adjointness supplies exactly that missing orthogonality, so the identities $P^2=P$ and $P^*=P$ become a complete intrinsic test for orthogonal projection.

[quotetheorem:9185]

This result lets the reader move freely between geometry and operator theory. When studying [Self-Adjoint Operators](/page/Self-Adjoint%20Operators), an orthogonal projection is the simplest nontrivial example: its spectrum is contained in $\{0,1\}$, and its range and kernel are perpendicular. The next characterisation explains why the same operator is also the natural solution to approximation problems. Once a closed subspace is interpreted as a class of allowable approximants, the projection is the element of that class nearest to the original vector.

[quotetheorem:9186]