# Orthonormal basis
An orthonormal basis is a basis of an inner product space whose vectors all have unit length and are mutually orthogonal, so that any vector in the space can be written as a linear combination of them with coefficients obtained by taking inner products. In a finite-dimensional real or complex inner product space, an orthonormal basis is a set \(\{\mathbf{q}_1, \ldots, \mathbf{q}_n\}\) with \(\langle \mathbf{q}_i, \mathbf{q}_j \rangle = \delta_{ij}\), where \(\delta_{ij}\) is the [Kronecker delta](https://www.edgepedia.org/kronecker-delta).[1](https://courses.math.wichita.edu/math511/ch5/3/OrthonormalBases.html) The concept extends to Hilbert spaces, the complete infinite-dimensional inner product spaces used in functional analysis and quantum mechanics.[2](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/pages/lecture-notes-and-readings/)
| Key fact | Detail |
|---|---|
| Defining property | Basis vectors satisfy \(\langle \mathbf{q}_i, \mathbf{q}_j \rangle = \delta_{ij}\): unit length and pairwise orthogonality[1](https://courses.math.wichita.edu/math511/ch5/3/OrthonormalBases.html) |
| Coordinates | For \(\mathbf{v} = \sum_i c_i \mathbf{q}_i\), the coefficients are \(c_i = \langle \mathbf{v}, \mathbf{q}_i \rangle\), so no linear system needs solving[1](https://courses.math.wichita.edu/math511/ch5/3/OrthonormalBases.html) • [3](https://ee.iitm.ac.in/~uday/notes/linalg/orth.html) |
| Parseval's identity | \(\lVert \mathbf{x} \rVert^2 = \sum_{i=1}^n x_i^2\) for coordinates in any orthonormal basis; it also holds in infinite-dimensional spaces[1](https://courses.math.wichita.edu/math511/ch5/3/OrthonormalBases.html) |
| Construction | Gram–Schmidt orthonormalization turns any independent set into an orthonormal basis of the same span, never dividing by zero when the vectors are independent[3](https://ee.iitm.ac.in/~uday/notes/linalg/orth.html) |
| Change of basis | A map taking one orthonormal basis to another is a unitary operator, satisfying \(\widehat{U}^{\dagger} = \widehat{U}^{-1}\)[4](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S8.html) |
| Infinite dimension | Any infinite-dimensional separable Hilbert space over the complex numbers is isomorphic to \(\ell^2\), the space of square-summable sequences[5](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/3d4cc88026d44a01f936cd6a0aa995cb_MIT18_102s20_lec_FA.pdf) |
| Fourier example | The normalized exponentials form an orthonormal basis for \(L^2\)[2](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/pages/lecture-notes-and-readings/) |
## Coordinates and the inner product
The practical advantage of an orthonormal basis is that coordinates become inner products. If \(\mathbf{v} = \sum_{i=1}^n c_i \mathbf{q}_i\), then taking the inner product with \(\mathbf{q}_j\) gives \(\langle \mathbf{v}, \mathbf{q}_j \rangle = c_j\), because every term \(\langle \mathbf{q}_i, \mathbf{q}_j \rangle\) vanishes except when \(i = j\).[1](https://courses.math.wichita.edu/math511/ch5/3/OrthonormalBases.html) With an arbitrary basis, finding coordinates means solving a linear system; with an orthonormal basis it means taking inner products.[3](https://ee.iitm.ac.in/~uday/notes/linalg/orth.html)
The inner product itself takes a simple diagonal form in such a basis. If \(\mathbf{x} = \sum_i x_i \mathbf{q}_i\) and \(\mathbf{y} = \sum_i y_i \mathbf{q}_i\), then \(\langle \mathbf{x}, \mathbf{y} \rangle = \sum_{i=1}^n x_i y_i\), the double sum collapsing through the Kronecker delta.[1](https://courses.math.wichita.edu/math511/ch5/3/OrthonormalBases.html) Geometrically, a vector is the sum of its projections onto the axes of the basis, and the rank-one projection matrices \(q_i q_i^T\) add up to the identity.[3](https://ee.iitm.ac.in/~uday/notes/linalg/orth.html)
**Parseval's identity** follows directly: for \(\mathbf{x} = \sum_{i=1}^n x_i \mathbf{q}_i\),
\[ \lVert \mathbf{x} \rVert^2 = \langle \mathbf{x}, \mathbf{x} \rangle = \sum_{i=1}^n x_i^2. \]
The identity holds in every orthonormal basis, not only the standard one, and it also holds for infinite-dimensional inner product spaces.[1](https://courses.math.wichita.edu/math511/ch5/3/OrthonormalBases.html) • [3](https://ee.iitm.ac.in/~uday/notes/linalg/orth.html) If the basis consists of sines and cosines instead, that identical result corresponds to the energy theorem in [Fourier analysis](https://www.edgepedia.org/fourier-analysis).[3](https://ee.iitm.ac.in/~uday/notes/linalg/orth.html)
## Constructing orthonormal bases
The [Gram–Schmidt process](https://www.edgepedia.org/gram-schmidt-process) converts any independent set \(a_1, \ldots, a_n\) into an orthonormal set spanning the same subspaces. The rule is to take the next given vector, subtract its components along everything already built, and normalize what is left:
\[ a_j' = a_j - \sum_{i=1}^{j-1} (q_i^T a_j)\, q_i, \qquad q_j = \frac{a_j'}{\lVert a_j' \rVert}. \]
The subtracted sum is exactly the projection of \(a_j\) onto the span of the earlier \(q_i\), so the leftover is perpendicular to all of them. If the original vectors are independent, the process never divides by zero, and \(\operatorname{span}(q_1, \ldots, q_k) = \operatorname{span}(a_1, \ldots, a_k)\) for every \(k\).[3](https://ee.iitm.ac.in/~uday/notes/linalg/orth.html)
Orthonormality also simplifies least squares. A matrix \(Q\) with orthonormal columns satisfies \(Q^T Q = I\), so the least squares solution for \(Qx = b\) collapses to \(\hat{x} = Q^T b\): no inverse to compute, just \(n\) inner products. A tall \(Q\) with \(m > n\) rows still has orthonormal columns, but \(QQ^T\) is then the projection onto the \(n\)-dimensional column space rather than the identity, and the rows are not orthonormal.[3](https://ee.iitm.ac.in/~uday/notes/linalg/orth.html)
## Orthonormal bases in Hilbert spaces
In a [Hilbert space](https://www.edgepedia.org/hilbert-space), an orthonormal basis is a maximal orthonormal sequence. For any element \(u\) of the space, the Fourier–Bessel series in such a basis converges to \(u\); if \(u - w\) is orthogonal to every basis vector, completeness of the basis forces \(u - w\) to vanish.[5](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/3d4cc88026d44a01f936cd6a0aa995cb_MIT18_102s20_lec_FA.pdf) Given a Hilbert space \(H\) with orthonormal basis \((e_i)_{i \in I}\), its Hilbert-space dimension is defined as the cardinality of the index set \(I\).[6](https://www.math.uni-sb.de/ag/speicher/weber/ISem24/ISem24LectureNotes.pdf)
A classification result states that there is only one infinite-dimensional separable Hilbert space up to isomorphism: any such complex Hilbert space is isomorphic to \(\ell^2\), the space of square-summable sequences.[2](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/pages/lecture-notes-and-readings/) • [5](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/3d4cc88026d44a01f936cd6a0aa995cb_MIT18_102s20_lec_FA.pdf) Concrete examples carry the theory: the normalized exponentials \(f_n(x) = \frac{1}{\sqrt{L}} \exp(2\pi i n x / L)\) form a complete orthonormal set of functions over the interval \([-L/2, L/2]\), and in the limit \(L \to \infty\) the corresponding sum passes into the integral representation of the [Dirac delta function](https://www.edgepedia.org/dirac-delta-function).[7](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf)
## Change of basis and quantum mechanics
In quantum mechanics, states live in a complex Hilbert space, and orthonormal bases are the frames in which measurement outcomes are read. A change from one orthonormal basis \(|n\rangle\) to another \(|\tilde{n}\rangle\) is written \(|\tilde{n}\rangle = \widehat{U}|n\rangle\), where \(\widehat{U}\) is a unitary operator, the analog of an orthogonal matrix rotating a coordinate system.[4](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S8.html) The new basis is orthonormal because \((U a_i | U a_j) = (a_i | U^{\dagger} U | a_j) = \delta_{ij}\).[7](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf)
Two operator classes are tied to orthonormal bases. Hermitian operators satisfy \(\widehat{H} = \widehat{H}^{\dagger}\) and have real eigenvalues; unitary operators satisfy \(\widehat{U}^{\dagger} = \widehat{U}^{-1}\) and have eigenvalues of modulus one. In finite-dimensional Hilbert spaces, Hermitian and unitary operators admit orthonormal bases of eigenvectors; in infinite-dimensional Hilbert spaces, this need not hold.[4](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S8.html) • [8](https://socr.umich.edu/TCIU/HTMLs/Chapter3_Radon_NikodymDerivatives.html) A common form writes a unitary operator as \(\widehat{U} = \exp(i \tau \widehat{H})\) for a hermitian \(\widehat{H}\) and real \(\tau\), in which case \(\widehat{H}\) and \(\widehat{U}\) share the same eigenvectors, with eigenvalues \(u_n = \exp(i \tau h_n)\).[4](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S8.html)
The position and momentum states of a particle illustrate a limiting case. The states \(|x\rangle\) for all \(x \in \mathbb{R}\) form a non-denumerable basis, one that cannot be labeled \(|1\rangle, |2\rangle, \ldots\). Their inner product is \((x|y) = \delta(x - y)\), so distinct position states are orthogonal, but the norm of a position state is infinite, \((x|x) = \delta(0)\), meaning these are idealizations rather than allowed particle states. Normalizable states are built as superpositions of them, using the completeness relation \(1 = \int dx\, |x\rangle\langle x|\).[7](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf)
## References
1. [Orthonormal Bases and Coordinates, MATH 511 notes, Wichita State University](https://courses.math.wichita.edu/math511/ch5/3/OrthonormalBases.html)
2. [Lecture Notes and Readings, 18.102 Introduction to Functional Analysis, MIT OpenCourseWare](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/pages/lecture-notes-and-readings/)
3. [EE5120 Linear Algebra: Orthogonality, IIT Madras](https://ee.iitm.ac.in/~uday/notes/linalg/orth.html)
4. [Mathematical interlude: Hilbert spaces and linear operators, Quantum Mechanics lecture notes, Lancaster University](https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S8.html)
5. [Functional Analysis Lecture Notes, Spring 2020, MIT OpenCourseWare](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/3d4cc88026d44a01f936cd6a0aa995cb_MIT18_102s20_lec_FA.pdf)
6. [ISEM24 Lecture Notes, Universität des Saarlandes](https://www.math.uni-sb.de/ag/speicher/weber/ISem24/ISem24LectureNotes.pdf)
7. [Quantum Physics II, Lecture Notes 4, MIT OpenCourseWare](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf)
8. [Spacekime Analytics (Time Complexity and Inferential Uncertainty)](https://socr.umich.edu/TCIU/HTMLs/Chapter3_Radon_NikodymDerivatives.html)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum formalism and states › Quantum states and wave functions › State vectors and Hilbert-space states › Bases and completeness relations*
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