Hilbert Space

A Hilbert space is a real or complex vector space equipped with an inner product and complete in the norm induced by that inner product. Quantum mechanics normally uses complex Hilbert spaces. Pure physical states are represented by rays in the space, while observables and dynamics are represented by suitable operators acting on it.

In Symbols

Let $\mathcal H$ be a complex vector space. In the physicists’ convention used here, the inner product $\langle\phi\vert\psi\rangle$ is conjugate-linear in its first argument and linear in its second:

$$\begin{aligned} \langle\phi\vert (\alpha\psi+\beta\chi)\rangle &=\alpha\langle\phi\vert\psi\rangle +\beta\langle\phi\vert\chi\rangle,\\ \langle (\alpha\phi+\beta\chi)\vert\psi\rangle &=\alpha^*\langle\phi\vert\psi\rangle +\beta^*\langle\chi\vert\psi\rangle. \end{aligned}$$

It also satisfies conjugate symmetry and positive definiteness:

$$\langle\phi\vert\psi\rangle =\langle\psi\vert\phi\rangle^*,$$ $$\langle\psi\vert\psi\rangle\ge0, \qquad \langle\psi\vert\psi\rangle=0 \iff \psi=0.$$

The induced norm and distance are

$$\lVert\psi\rVert =\sqrt{\langle\psi\vert\psi\rangle}, \qquad d(\phi,\psi)=\lVert\phi-\psi\rVert.$$

Completeness means that every Cauchy sequence in this norm converges to a vector that remains inside $\mathcal H$:

$$\forall\varepsilon>0,\ \exists N:\ m,n>N \Longrightarrow \lVert\psi_m-\psi_n\rVert<\varepsilon,$$

implies that some $\psi\in\mathcal H$ satisfies

$$\lVert\psi_n-\psi\rVert\longrightarrow0.$$

Completeness is the property that turns an inner-product space into a Hilbert space.

Canonical Home

See Hilbert Spaces for the canonical mathematical treatment. L2 Spaces develops square-integrable function spaces, while State Vectors explains how Hilbert-space vectors enter quantum formalism.

The choice of which inner-product argument is linear varies between physics and parts of mathematics. Inner-Product Conventions gives the translation rule.

Standard Examples

Finite-Dimensional Space

The space $\mathbb C^n$ with

$$\langle\phi\vert\psi\rangle =\sum_{j=1}^n\phi_j^*\psi_j$$

is a Hilbert space. Every finite-dimensional inner-product space is automatically complete.

A qubit uses $\mathcal H=\mathbb C^2$. A normalized vector can be written

$$\lvert\psi\rangle =\alpha\lvert0\rangle+\beta\lvert1\rangle, \qquad \lvert\alpha\rvert^2+\lvert\beta\rvert^2=1.$$

Square-Integrable Functions

For a measure space $(X,\mu)$,

$$L^2(X,d\mu) =\left\lbrace \psi: \int_X\lvert\psi(x)\rvert^2\,d\mu(x)<\infty \right\rbrace$$

is a Hilbert space after functions that agree almost everywhere are identified. Its inner product is

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

The almost-everywhere identification matters: elements of $L^2$ are equivalence classes, not functions with intrinsically defined values at every point.

Composite Systems

If systems $A$ and $B$ have Hilbert spaces $\mathcal H_A$ and $\mathcal H_B$, their composite system uses the completed tensor product

$$\mathcal H_{AB} =\mathcal H_A\otimes\mathcal H_B.$$

The completion allows limits of finite sums of product vectors. In finite dimensions this extra word changes nothing, but in infinite dimensions it is part of the construction.

Direct sums,

$$\mathcal H=\mathcal H_1\oplus\mathcal H_2,$$

instead describe alternatives or sectors rather than simultaneous subsystems. Tensor product and direct sum solve different structural problems.

Rays and Normalization

A nonzero vector and any nonzero scalar multiple of it define the same ray:

$$\lvert\psi\rangle \sim c\lvert\psi\rangle, \qquad c\in\mathbb C\setminus\{0\}.$$

Normalized representatives satisfy

$$\langle\psi\vert\psi\rangle=1,$$

and then differ only by a global phase $e^{i\alpha}$. The physical pure-state space is therefore projective Hilbert space, not the unit sphere with distinct global phases.

The zero vector is never a physical pure state because it cannot be normalized and defines no ray.

Bases and Expansions

An orthonormal set $\{\lvert n\rangle\}$ obeys

$$\langle m\vert n\rangle=\delta_{mn}.$$

It is a complete orthonormal basis when every vector admits a norm-convergent expansion

$$\lvert\psi\rangle =\sum_n \lvert n\rangle\langle n\vert\psi\rangle,$$

with Parseval’s identity

$$\lVert\psi\rVert^2 =\sum_n \lvert\langle n\vert\psi\rangle\rvert^2.$$

In a separable Hilbert space, a countable orthonormal basis exists. Separable does not mean finite-dimensional.

Formal position kets $\lvert x\rangle$ satisfy delta normalization rather than unit normalization:

$$\langle x\vert x'\rangle=\delta(x-x').$$

They are generalized eigenvectors, not ordinary members of $L^2(\mathbb R)$. A continuous spectral resolution is not an uncountable orthonormal Hilbert basis in the same sense as the countable sum above. Generalized Eigenvectors develops this distinction.

Closed Subspaces and Projection

A linear subspace $\mathcal M\subseteq\mathcal H$ is closed if it contains all its norm limits. A closed subspace is itself a Hilbert space with the inherited inner product.

Every closed subspace has an orthogonal complement

$$\mathcal M^\perp =\left\lbrace \psi\in\mathcal H: \langle\phi\vert\psi\rangle=0 \text{ for all }\phi\in\mathcal M \right\rbrace,$$

and

$$\mathcal H=\mathcal M\oplus\mathcal M^\perp.$$

The orthogonal projector $P_{\mathcal M}$ onto $\mathcal M$ satisfies

$$P_{\mathcal M}^2=P_{\mathcal M}, \qquad P_{\mathcal M}^\dagger=P_{\mathcal M}.$$

Closedness is what guarantees that the nearest-point orthogonal projection lies in the subspace.

Operators and Domains

A bounded linear operator on a Hilbert space is defined everywhere and continuous. Important quantum operators such as position, momentum, and most Hamiltonians are unbounded. An unbounded operator is specified by both its action and a dense domain:

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

Two differential operators with the same formal expression but different boundary conditions can be different operators with different spectra. Adjointness, self-adjointness, products, and commutators depend on domains.

Unbounded Operators and Hermitian versus Self-Adjoint are the canonical references for these cautions.

Finite and Infinite Dimensions

Several shortcuts valid in finite dimensions fail in infinite dimensions:

Finite-dimensional fact	Infinite-dimensional caution
Every linear operator is bounded	Important operators can be unbounded
Every subspace is closed	A subspace can be dense but not closed
Every symmetric matrix is self-adjoint	A symmetric operator can have several self-adjoint extensions or none
Every bounded sequence has a convergent subsequence	Bounded sets need not be compact
Every spectrum consists of eigenvalues	Continuous and residual spectra can occur

The abstract Hilbert-space framework covers both cases, but the analytic work is much heavier in infinite dimensions.

Common Aliases

• complete inner-product space;
• state space, when context makes clear that rays or density operators represent physical states;
• ket space, informally;
• physical Hilbert space, when constraints have already been imposed.

The phrase state space can also mean a convex set of density operators or a classical phase space, so context is essential.

Common Confusions

• A Hilbert space is not necessarily finite-dimensional.
• An inner-product space need not be complete.
• Vectors are not the same as their coordinate columns in one chosen basis.
• Normalized vectors differing by a global phase represent the same pure state.
• The zero vector is not a quantum state.
• A dense subspace need not be closed or complete in the inherited norm.
• Generalized position eigenstates are not ordinary elements of $L^2(\mathbb R)$.
• A continuous generalized basis is not an ordinary Hilbert basis.
• A direct sum is not a tensor product.
• A Hilbert space alone does not specify the Hamiltonian, observables, domains, or physical interpretation of a theory.
• Operator domains matter in infinite-dimensional Hilbert spaces.

Related Entries

• Wavefunction
• Observable
• Density Matrix
• Superposition
• Tensor Products
• Projectors
• Rays and Global Phase

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, chs. 1 and 4.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, ch. 1.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013, chs. 2–4.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, rev. and enlarged ed., Academic Press, 1980, chs. I–VIII.