Infinite-Dimensional Hilbert Spaces

Infinite-dimensional Hilbert spaces are the natural setting for wavefunctions, differential operators, continuous spectra, and scattering states. They retain the geometry of inner products and orthogonal projection, but several conveniences of finite-dimensional linear algebra disappear. Completeness is no longer automatic, linear operators need not be bounded or everywhere defined, and formal eigenkets may lie outside the Hilbert space.

This chapter is a bridge from matrices to operator theory. Its guiding rule is simple: in infinite dimension, the Hilbert space, operator action, operator domain, convergence mode, and spectral interpretation are all part of the statement. Suppressing any one of them can turn a valid calculation into a false claim.

What changes in infinite dimension

Finite-dimensional habit	Infinite-dimensional correction
Every inner-product space is complete	completeness must be required or obtained by completion
Every linear map is bounded	important quantum operators such as position, momentum, and Hamiltonians are often unbounded
An operator is specified by a matrix formula	an unbounded operator is specified by both its action and its domain
Hermitian and self-adjoint are interchangeable	symmetry and self-adjointness differ because adjoint domains can be larger
A spectral decomposition is a finite or countable eigenvector sum	continuous spectral parts require projection-valued measures and integrals
Every spectral ket is a vector in the state space	$\lvert x\rangle$ and $\lvert p\rangle$ are generalized vectors or distributions
Every operator has a finite trace	traces require trace-class hypotheses

Finite-dimensional intuition remains useful, but only after the missing hypotheses are restored.

The state-space layer

A Hilbert space $\mathcal H$ is an inner-product space complete in the induced norm

$$\lVert\psi\rVert =\sqrt{\langle\psi,\psi\rangle}.$$

Completeness means that every Cauchy sequence in this norm converges to an element of $\mathcal H$. It guarantees that limits of physically meaningful approximation procedures do not leave the state space.

The central wave-mechanics example is

$$L^2(X,\mu) =\left\{ \psi: \int_X\lvert\psi(x)\rvert^2\,d\mu(x)\lt\infty \right\} \big/\text{equality almost everywhere},$$

with inner product

$$\langle\phi,\psi\rangle =\int_X\phi(x)^*\psi(x)\,d\mu(x).$$

An $L^2$ vector is an equivalence class of functions, not a pointwise-defined function. Changing a representative on a set of measure zero does not change the Hilbert-space vector. Consequently, point evaluation is not intrinsically defined on general $L^2$ states.

An orthonormal Hilbert basis $\{e_n\}$ satisfies

$$\psi=\sum_{n=1}^{\infty} \langle e_n,\psi\rangle e_n,$$

where the series converges in norm. Norm convergence does not by itself imply pointwise or uniform convergence of function representatives. A Hilbert space is separable when it has a countable dense subset, equivalently a countable orthonormal basis when nonzero. Most standard single-particle spaces used in quantum mechanics are separable, despite having uncountably many vectors.

Use Hilbert Spaces, $L^2$ Spaces, Completeness and Orthonormal Bases, and Separable Hilbert Spaces for this layer.

Operators are action plus domain

A general operator is written

$$A:\mathcal D(A)\subseteq\mathcal H\longrightarrow\mathcal H.$$

For a bounded operator defined on all of $\mathcal H$, there is a constant $C$ such that

$$\lVert A\psi\rVert \leq C\lVert\psi\rVert$$

for every $\psi\in\mathcal H$. The least such $C$ is the operator norm. Boundedness is equivalent to continuity, and a bounded operator defined on a dense subspace extends uniquely to the whole Hilbert space.

An unbounded operator cannot be defined everywhere while retaining the standard closed-operator framework. In particular, the Hellinger–Toeplitz theorem says that an everywhere-defined symmetric operator on a Hilbert space is bounded. Differential operators therefore come with proper dense domains encoding differentiability, integrability, and boundary conditions.

Two expressions can have the same differential action but define different operators:

$$(A\psi)(x)=a(x)\psi(x) \quad\text{or}\quad (A\psi)(x)=-i\hbar\psi'(x)$$

becomes an operator only after $\mathcal D(A)$ is stated. This is why Unbounded Operators and Domains of Operators should be read together.

Adjoint, symmetric, and self-adjoint

Let $A$ be densely defined. With the physics convention for the inner product, a vector $\phi$ lies in $\mathcal D(A^\dagger)$ when there exists a vector $\eta\in\mathcal H$ such that

$$\langle\phi,A\psi\rangle =\langle\eta,\psi\rangle \quad \text{for every }\psi\in\mathcal D(A).$$

The vector $\eta$ is unique, and $A^\dagger\phi=\eta$. Thus the adjoint’s domain is determined by a boundedness condition on the inner-product pairing; it is not obtained merely by conjugating a differential expression.

The domain distinctions are

$$\begin{aligned} A\text{ symmetric} &\quad\Longleftrightarrow\quad \mathcal D(A)\subseteq\mathcal D(A^\dagger), \quad A^\dagger\psi=A\psi \text{ on }\mathcal D(A),\\ A\text{ self-adjoint} &\quad\Longleftrightarrow\quad \mathcal D(A)=\mathcal D(A^\dagger), \quad A=A^\dagger. \end{aligned}$$

Self-adjointness is stronger. It supplies the spectral theorem and, through Stone’s theorem, unitary evolution generated by the operator. Read Adjoint Operators before Symmetric versus Self-Adjoint Operators.

Boundary conditions are operator data

Consider the momentum differential expression on $L^2([0,L])$,

$$P=-i\hbar\frac{d}{dx}.$$

Integration by parts gives the boundary form

$$\langle\phi,P\psi\rangle -\langle P\phi,\psi\rangle =-i\hbar \left[\phi(x)^*\psi(x)\right]_{0}^{L}.$$

The action is symmetric only on domains where this expression vanishes for every pair of domain vectors. The quasi-periodic family

$$\psi(L)=e^{i\theta}\psi(0), \qquad 0\leq\theta\lt2\pi,$$

with the corresponding Sobolev regularity, gives self-adjoint momentum operators with different spectra. The boundary condition is therefore not an optional instruction attached after solving the eigenvalue equation; it helps define which operator is being studied. The canonical general treatment is Boundary Conditions.

The spectral-measure layer

For a self-adjoint operator $A$, the spectral theorem provides a projection-valued measure $E_A$ on the real line such that

$$A=\int_{\mathbb R}\lambda\,dE_A(\lambda).$$

For suitable functions $f$,

$$f(A)=\int_{\mathbb R}f(\lambda)\,dE_A(\lambda).$$

Given a normalized state $\psi$, the scalar measure

$$\mu_\psi(\Delta) =\langle\psi,E_A(\Delta)\psi\rangle$$

is a probability measure on spectral outcomes. Discrete spectral sums are the special case in which $E_A$ is concentrated on eigenvalues. Continuous spectrum does not mean that ordinary normalized eigenvectors suddenly form an uncountable Hilbert basis.

For an unbounded function of $A$, the domain is part of the functional calculus:

$$\mathcal D(f(A)) =\left\{ \psi\in\mathcal H: \int_{\mathbb R} \lvert f(\lambda)\rvert^2 \,d\mu_\psi(\lambda) \lt\infty \right\}.$$

The Spectral Theorem, Practical Version develops this working language. Continuous Spectra explains how it differs from a discrete eigensystem.

Representations and generalized eigenvectors

Position and momentum wavefunctions are coordinate representations of one abstract state:

$$\psi(x)=\langle x|\psi\rangle, \qquad \widetilde\psi(p)=\langle p|\psi\rangle.$$

With a common Fourier convention,

$$\widetilde\psi(p) =\frac{1}{\sqrt{2\pi\hbar}} \int_{\mathbb R} e^{-ipx/\hbar}\psi(x)\,dx.$$

Plancherel’s theorem makes this transform unitary on $L^2(\mathbb R)$, even though the integral formula may first be defined on a dense, better-behaved class of functions.

The symbols $|x\rangle$ and $|p\rangle$ are not normalizable vectors in $L^2(\mathbb R)$. Their formal relations

$$\langle x|x'\rangle=\delta(x-x'), \qquad I=\int_{\mathbb R}|x\rangle\langle x|\,dx$$

are distributional statements. A rigged Hilbert space places them in a continuous dual or antidual:

$$\Phi\subset\mathcal H\subset\Phi^\times.$$

Test vectors lie in $\Phi$, normalizable states lie in $\mathcal H$, and generalized kets act on test vectors through $\Phi^\times$. Use Position and Momentum Representations, Generalized Eigenvectors, and Rigged Hilbert Spaces, First Look in that order. Distribution theory itself is canonical in Distributions.

Trace ideals

On an infinite-dimensional Hilbert space, boundedness does not guarantee a finite trace. If $s_n(T)$ are the singular values of a compact operator $T$, then

$$\begin{aligned} T\text{ is Hilbert--Schmidt} &\quad\Longleftrightarrow\quad \sum_n s_n(T)^2\lt\infty,\\ T\text{ is trace class} &\quad\Longleftrightarrow\quad \sum_n s_n(T)\lt\infty. \end{aligned}$$

The inclusions are

$$\mathcal S_1(\mathcal H) \subset\mathcal S_2(\mathcal H) \subset\mathcal K(\mathcal H) \subset\mathcal B(\mathcal H),$$

where $\mathcal K$ denotes compact operators and $\mathcal B$ bounded operators. Density operators are positive trace-class operators of trace one. Trace-Class and Hilbert-Schmidt Operators is the canonical bridge to density operators and partial traces.

Page map

Page	Central question
Hilbert Spaces	Why must an inner-product state space be complete?
$L^2$ Spaces	What does square integrability mean, and why are wavefunctions equivalence classes?
Completeness and Orthonormal Bases	In which sense does an infinite basis expansion converge?
Separable Hilbert Spaces	Why can a countable basis describe standard infinite-dimensional state spaces?
Bounded Operators	Which operators are continuous and defined safely on the full space?
Unbounded Operators	Why do position, momentum, and Hamiltonians require additional care?
Domains of Operators	How do regularity and boundary conditions become part of an operator?
Adjoint Operators	How does the inner product determine an adjoint and its domain?
Symmetric versus Self-Adjoint Operators	Why is symmetry weaker than self-adjointness?
Spectral Theorem, Practical Version	How do projection-valued measures unify discrete and continuous spectra?
Continuous Spectra	What replaces a normalizable eigenbasis in continuous spectral sectors?
Position and Momentum Representations	How does one state acquire position- and momentum-space wavefunctions?
Generalized Eigenvectors	How should delta-normalized kets be read and used?
Rigged Hilbert Spaces, First Look	Which larger space gives generalized kets a disciplined home?
Trace-Class and Hilbert-Schmidt Operators	Which operator classes make traces and density operators controlled?

Reading routes

• Wave mechanics: Hilbert spaces $\to$ $L^2$ spaces $\to$ complete orthonormal systems $\to$ position and momentum representations $\to$ continuous spectra.
• Observable operators: bounded operators $\to$ unbounded operators $\to$ domains $\to$ adjoints $\to$ self-adjointness $\to$ the spectral theorem.
• Scattering and generalized states: continuous spectra $\to$ generalized eigenvectors $\to$ rigged Hilbert spaces, followed by Scattering Amplitude.
• Density operators: bounded operators $\to$ adjoints $\to$ trace ideals, followed by Density Operators.
• Rigorous foundations: complete the operator route, then use the Math Needed for Core Formalism crosswalk and the rigorous-QM references below.

Common mistakes

Mistake	Correction
Treating an $L^2$ vector as a uniquely defined pointwise function	remember that representatives equal almost everywhere define one vector
Writing an unbounded operator without its domain	state both the action and $\mathcal D(A)$
Computing a formal conjugate transpose and calling it the adjoint	determine the adjoint domain from the inner-product identity
Showing only that an operator is symmetric	self-adjointness also requires equality of domains
Treating boundary conditions as external to the operator	include them in the domain; different choices may give different spectra
Calling $\lvert x\rangle$ a normalized Hilbert-space vector	interpret it as a generalized eigenvector or distribution
Replacing norm convergence by pointwise convergence	name and verify the convergence mode actually needed
Taking traces of arbitrary bounded operators	establish trace-class conditions first

Exercises

1. Equality almost everywhere

Let $f(x)=0$ for all $x\in[0,1]$, and let $g$ equal zero except that $g(1/2)=7$. What is $\lVert f-g\rVert_{L^2}$? Are $f$ and $g$ different Hilbert-space vectors?

Solution

The functions differ only on a one-point set of measure zero, so

$$\lVert f-g\rVert_{L^2}^2 =\int_0^1\lvert f(x)-g(x)\rvert^2\,dx =0.$$

They are different pointwise representatives of the same element of $L^2([0,1])$. This is why point evaluation is not a well-defined operation on an arbitrary $L^2$ equivalence class.

2. The position operator is unbounded

On $L^2(\mathbb R)$, let $(X\psi)(x)=x\psi(x)$. Define $\psi_n$ as the normalized indicator function of $[n,n+1]$. Show that $\lVert\psi_n\rVert=1$ while $\lVert X\psi_n\rVert\geq n$.

Solution

The interval has length one, so $\lVert\psi_n\rVert^2=1$. Moreover,

$$\lVert X\psi_n\rVert^2 =\int_n^{n+1}x^2\,dx \geq n^2.$$

Hence $\lVert X\psi_n\rVert\geq n$. No finite constant $C$ can satisfy $\lVert X\psi\rVert\leq C\lVert\psi\rVert$ on the operator’s domain, so $X$ is unbounded.

3. Quasi-periodic momentum boundary form

Suppose $\phi(L)=e^{i\theta}\phi(0)$ and $\psi(L)=e^{i\theta}\psi(0)$. Show that the momentum boundary form vanishes.

Solution

At the upper endpoint,

$$\phi(L)^*\psi(L) =e^{-i\theta}\phi(0)^* e^{i\theta}\psi(0) =\phi(0)^*\psi(0).$$

Therefore $[\phi^*\psi]_0^L=0$, so $\langle\phi,P\psi\rangle=\langle P\phi,\psi\rangle$ on this domain. This proves symmetry. Establishing self-adjointness additionally requires checking that the adjoint has the same boundary-condition domain.

4. Spectral probability measure

Let $E_A$ be the projection-valued measure of a self-adjoint operator and let $\lVert\psi\rVert=1$. Show that $\mu_\psi(\Delta)=\langle\psi,E_A(\Delta)\psi\rangle$ is normalized and nonnegative.

Solution

Every $E_A(\Delta)$ is an orthogonal projector, so

$$\mu_\psi(\Delta) =\lVert E_A(\Delta)\psi\rVert^2 \geq0.$$

Because $E_A(\mathbb R)=I$,

$$\mu_\psi(\mathbb R) =\langle\psi,I\psi\rangle =1.$$

Countable additivity follows from the corresponding projection-valued-measure property, with orthogonal projections on disjoint sets.

References

• J. B. Conway, A Course in Functional Analysis, 2nd ed., Springer, 1990.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.
• B. Simon, Trace Ideals and Their Applications, 2nd ed., American Mathematical Society, 2005.
• G. Teschl, Mathematical Methods in Quantum Mechanics, 2nd ed., American Mathematical Society, 2014.
• J. Weidmann, Linear Operators in Hilbert Spaces, Springer, 1980.