# Square-integrable function

A square-integrable function is a function whose absolute value squared has a finite integral over its domain, with the integral understood in the sense of Lebesgue. On a measure space \( (X, \mu) \), a function \( f: X \to \mathbb{R} \) is square integrable when \( \int_X |f|^2 \, d\mu \) exists and is finite, and \( \sqrt{\int_X |f|^2 \, d\mu} \) is then its \( L^2 \)-norm.<sup>[1](https://mathworld.wolfram.com/L2-Function.html)</sup> The collection of such functions on \( X \) is written \( L^2(X) \), pronounced "ell-two", and forms a [Hilbert space](https://www.edgepedia.org/hilbert-space), a complete vector space with an inner product.<sup>[1](https://mathworld.wolfram.com/L2-Function.html)</sup> The definition extends to complex-valued functions by replacing the absolute value of a real number with the modulus (norm) of a complex number, and further to functions taking values in any normed space.<sup>[1](https://mathworld.wolfram.com/L2-Function.html)</sup>

| Key fact | Detail |
|---|---|
| Defining condition | \( \int_X \|f\|^2 \, d\mu < \infty \), with the Lebesgue integral on a measure space \( (X, \mu) \)<sup>[1](https://mathworld.wolfram.com/L2-Function.html)</sup> |
| The space \( L^2(X) \) | The set of square-integrable functions on \( X \); it is a Hilbert space<sup>[1](https://mathworld.wolfram.com/L2-Function.html)</sup> |
| Pointwise meaning | Elements of \( L^2(\mathbb{R}) \) are strictly equivalence classes of functions agreeing almost everywhere; \( \psi \in L^2(\mathbb{R}) \) is shorthand for \( [\psi] \in L^2(\mathbb{R}) \)<sup>[2](https://www.thp.uni-koeln.de/gross/files/qm.pdf)</sup> |
| Boundary example | On \( (0,1) \), \( f(x) = 1/x^p \) lies in \( L^2 \) for \( p < 1/2 \), while \( x^{-1/2} \) does not, since \( \int_0^1 (x^{-1/2})^2 \, dx = \int_0^1 dx/x \) diverges<sup>[1](https://mathworld.wolfram.com/L2-Function.html)</sup> |
| Fourier transform | By Plancherel's theorem (1910), the Fourier transform extends to a unitary operator on \( L_2 \), preserving the norm: \( \int \|f(x)\|^2 dx = \int \|\widehat{f}(\lambda)\|^2 d\lambda \)<sup>[3](https://encyclopediaofmath.org/index.php?title=Plancherel_theorem)</sup> |
| Role in quantum mechanics | A wavefunction is normalizable, or square-integrable, when \( \int \|\Psi\|^2 dx < \infty \); probabilities are given by integrals of the absolute square of \( \Psi \)<sup>[4](https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2016/d87c768c4e1181fde8d4366112169719_MIT8_04S16_LecNotes6.pdf)</sup><sup> • </sup><sup>[1](https://mathworld.wolfram.com/L2-Function.html)</sup> |

## The space L² and its norm

The norm on \( L^2 \) behaves like the length of a vector. For real-valued functions on an interval, \( \|f\| = \left( \int f^2 \, d\mu \right)^{1/2} \), and it satisfies \( \|af\| = \|a\| \, \|f\| \) for every scalar \( a \). The norm is nonnegative and vanishes only when \( f = 0 \) almost everywhere, that is, everywhere except on a set of measure zero.<sup>[5](https://citeseerx.ist.psu.edu/document?doi=f35cc004cf88909d896afdd5a199788d36d7dd28&repid=rep1&type=pdf)</sup>

**Equivalence classes.** Because functions that differ only on a set of measure zero have the same norm, they cannot be distinguished inside \( L^2 \). Rigorous treatments therefore define \( L^2(\mathbb{R}) \) as the set of equivalence classes \( [\psi] \) of functions \( \psi: \mathbb{R} \to \mathbb{C} \) with \( \int \|\psi(x)\|^2 dx < \infty \), where two functions belong to the same class when they agree almost everywhere. In practice the identification is left implicit, and \( L^2(\mathbb{R}) \) is called "the space of square-integrable functions" rather than the more precise "space of equivalence classes of square-integrable functions".<sup>[2](https://www.thp.uni-koeln.de/gross/files/qm.pdf)</sup> This convention has a physical counterpart in quantum mechanics: wave functions that agree almost everywhere define the same physical state and are identified.<sup>[2](https://www.thp.uni-koeln.de/gross/files/qm.pdf)</sup>

The inner product \( \langle \psi \mid \phi \rangle = \int \psi(x)^* \phi(x) \, dx \) is well defined on \( L^2 \) because the [Cauchy–Schwarz inequality](https://www.edgepedia.org/cauchy-schwarz-inequality) bounds it by the product of the two norms,<sup>[2](https://www.thp.uni-koeln.de/gross/files/qm.pdf)</sup>

\[ \left| \int \psi(x)^* \phi(x) \, dx \right| \leq \left( \int \|\psi(x)\|^2 dx \right)^{1/2} \left( \int \|\phi(x)\|^2 dx \right)^{1/2}. \]

## What membership in L² requires

Square integrability constrains how fast a function can grow or decay. On the unit interval \( (0,1) \), the functions \( f(x) = 1/x^p \) belong to \( L^2 \) exactly when \( p < 1/2 \); the borderline case \( f(x) = x^{-1/2} \) fails because its square is \( 1/x \), whose integral over \( (0,1) \) does not exist.<sup>[1](https://mathworld.wolfram.com/L2-Function.html)</sup> A function can also fail to have a well-defined limit at infinity and still be square integrable, so membership in \( L^2 \) is a statement about the integral of the squared magnitude, not about pointwise behavior at every point.<sup>[4](https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2016/d87c768c4e1181fde8d4366112169719_MIT8_04S16_LecNotes6.pdf)</sup>

Derivatives impose separate requirements. A generic function in \( L^2 \) does not admit derivatives that again lie in \( L^2 \), so differential operators such as the Laplacian are not defined on the whole space; their domains are proper subspaces such as the [Sobolev space](https://www.edgepedia.org/sobolev-space) \( H^2(\mathbb{R}^n) \), consisting of functions whose weak derivatives up to second order remain square integrable.<sup>[6](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/fackler/SS15/qm/lnotes_mathematical_found_qm_temp.pdf)</sup>

## The Fourier transform on L²

For a function \( f \in L_2(-\infty, +\infty) \), the truncated Fourier integrals \( \widehat{f}_\omega(x) = \frac{1}{\sqrt{2\pi}} \int_{-\omega}^{+\omega} f(y) e^{-ixy} \, dy \) converge in the \( L_2 \) norm as \( \omega \to \infty \) to a function \( \widehat{f} \in L_2 \), and \( f \) is likewise recovered from \( \widehat{f} \) as an \( L_2 \)-limit of inverse integrals.<sup>[3](https://encyclopediaofmath.org/index.php?title=Plancherel_theorem)</sup> The integrals exist in the sense of the principal value for almost all \( x \).<sup>[3](https://encyclopediaofmath.org/index.php?title=Plancherel_theorem)</sup>

This is the content of Plancherel's theorem, established by M. Plancherel in 1910: the [Fourier transform](https://www.edgepedia.org/fourier-transform) \( \mathcal{F} \) and its inverse \( \mathcal{F}^{-1} \) are mutually inverse unitary operators on \( L_2 \), and the transformation preserves the norm,<sup>[3](https://encyclopediaofmath.org/index.php?title=Plancherel_theorem)</sup>

\[ \int_{-\infty}^{+\infty} \|f(x)\|^2 \, dx = \int_{-\infty}^{+\infty} \|\widehat{f}(\lambda)\|^2 \, d\lambda, \]

the Parseval–Plancherel formula. The theorem extends from \( \mathbb{R} \) to \( \mathbb{R}^n \) and to locally compact Abelian groups, where the Fourier transform maps \( L_2(X) \) to \( L_2(Y) \) with \( Y \) the character group of \( X \).<sup>[3](https://encyclopediaofmath.org/index.php?title=Plancherel_theorem)</sup><sup> • </sup><sup>[7](https://encyclopediaofmath.org/wiki/Plancherel_formula)</sup> A related convergence result is due to [Lennart Carleson](https://www.edgepedia.org/lennart-carleson), who proved in 1966 that the [Fourier series](https://www.edgepedia.org/fourier-series) of any square-integrable function converges pointwise almost everywhere to the function.<sup>[8](https://www.math.uci.edu/node/37981)</sup>

## Square-integrable functions in quantum mechanics

In quantum mechanics, probabilities are given as integrals of the absolute square of a wavefunction \( \psi \), and \( L^2 \)-functions arise because these quantities, like energy densities, must remain finite.<sup>[1](https://mathworld.wolfram.com/L2-Function.html)</sup> A wavefunction with \( \int dx \, \|\Psi\|^2 = \mathcal{N} \neq 1 \) is normalizable: the probability of finding the particle in an interval \( dx \) about \( x \) is \( dP = \frac{1}{\mathcal{N}} \|\Psi\|^2 dx \), and multiplying \( \Psi \) by \( 1/\sqrt{\mathcal{N}} \) produces a properly normalized state with \( \int \|\Psi'\|^2 dx = 1 \). The overall scale of \( \Psi \) carries no physics.<sup>[4](https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2016/d87c768c4e1181fde8d4366112169719_MIT8_04S16_LecNotes6.pdf)</sup>

**Non-normalizable idealizations.** Some widely used states are not square integrable. The de Broglie plane wave \( \Psi = e^{ikx - i\omega t} \) has \( \|\Psi \|^2 = 1 \) everywhere, so its integral is infinite, and it does not truly represent a single particle; a square-integrable wavefunction can be built as a superposition of plane waves.<sup>[4](https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2016/d87c768c4e1181fde8d4366112169719_MIT8_04S16_LecNotes6.pdf)</sup> Similarly, the position eigenstate has wavefunction \( \langle x'' \mid x' \rangle = \delta(x'' - x') \), which is not a square-integrable function and is not an element of the Hilbert space, although all square-integrable functions can be built out of position states; momentum eigenstates behave the same way.<sup>[9](https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf)</sup> One way to make this precise is to change the problem: place the particle in a box of size \( L \) with periodic boundary conditions, which gives discrete momenta \( p' L = 2\pi n \) with \( n \in \mathbb{Z} \) and a countable basis of normalizable momentum eigenstates, then take \( L \) to infinity at the end.<sup>[9](https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf)</sup>

The position-space and momentum-space wavefunctions of any square-integrable state are related by the Fourier transform, and the Parseval identity expresses the equality of total probability computed in either representation.<sup>[9](https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf)</sup><sup> • </sup><sup>[10](https://openscienceteam.github.io/Quantum-Mechanics/hilbert/)</sup> Square integrability also underlies conservation of probability: for the free [Schrödinger equation](https://www.edgepedia.org/schrodinger-equation), unitary time evolution preserves the \( L^2 \) norm, \( \|\psi(t)\|_2 = \|\psi_0\|_2 \), so a normalized initial state keeps total probability equal to 1 for all time.<sup>[11](https://people.clas.ufl.edu/shabanov/files/MathPhysicsWSPC2025Chapter5.pdf)</sup>

## Orthonormal expansions

Like any Hilbert space, \( L^2 \) admits orthonormal bases. For a particle in a box of length \( L \), the functions \( \phi_n(x) = \sqrt{2/L} \, \sin(n\pi x / L) \) are orthonormal, and any continuous square-integrable wave function has a unique expansion \( \psi(x) = \sum_{n=1}^{\infty} c_n \phi_n(x) \), the content of Fourier's theorem, with coefficients \( c_n = \langle \phi_n, \psi \rangle \). The squared norm becomes \( \langle \psi, \psi \rangle = \int_0^L dx \, \|\psi(x)\|^2 = \sum_{n=1}^{\infty} \|c_n\|^2 \), which is Parseval's theorem.<sup>[12](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/hilbert_space.html)</sup>

## References

1. [L^2-Function, Wolfram MathWorld (Weisstein, Eric W., with contributions by Todd Rowland)](https://mathworld.wolfram.com/L2-Function.html)
2. [Mathematical Foundations of Quantum Mechanics, lecture notes, Universität zu Köln, Appendix D: Function Spaces and Distributions](https://www.thp.uni-koeln.de/gross/files/qm.pdf)
3. [Plancherel theorem, Encyclopedia of Mathematics (P.I. Lizorkin, originator)](https://encyclopediaofmath.org/index.php?title=Plancherel_theorem)
4. [Quantum Physics I, Lecture Note 6, MIT OpenCourseWare 8.04 (Spring 2016)](https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2016/d87c768c4e1181fde8d4366112169719_MIT8_04S16_LecNotes6.pdf)
5. [Notes on Measure and Integration, Chapter 5: The Hilbert Space L²[−1, 1], CiteSeerX](https://citeseerx.ist.psu.edu/document?doi=f35cc004cf88909d896afdd5a199788d36d7dd28&repid=rep1&type=pdf)
6. [Lecture Notes: Mathematical Foundations of Quantum Mechanics, Universität Ulm](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/fackler/SS15/qm/lnotes_mathematical_found_qm_temp.pdf)
7. [Plancherel formula, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Plancherel_formula)
8. [A degree one Carleson operator along the paraboloid, UCI Mathematics](https://www.math.uci.edu/node/37981)
9. [Quantum Theory I, Lecture 5 Notes, MIT OpenCourseWare 8.321 (Fall 2017)](https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf)
10. [The Mathematical Formalism of Quantum Mechanics, Open Science Team, Quantum Mechanics](https://openscienceteam.github.io/Quantum-Mechanics/hilbert/)
11. [Fundamental Solutions and the Schrödinger Equation, Chapter 5, University of Florida (S. Shabanov)](https://people.clas.ufl.edu/shabanov/files/MathPhysicsWSPC2025Chapter5.pdf)
12. [The Hilbert Space, lecture notes, Durham University (Kasper Peeters)](https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/hilbert_space.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 › Wave functions and position-space states › Normalizability and square-integrable wave functions*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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