nLab Hilbert space

Idea

Dr. von Neumann, ich möchte gerne wissen, was ist denn eigentlich ein Hilbertscher Raum ? ¹

A Hilbert space is (see Def. for details):

1. a (real or, usually, complex) vector space, possibly of infinite dimension,

2. equipped with a positive definite Hermitian inner product,

3. which, as a topological space, is complete with respect to the induced metric.

Hilbert spaces are central to quantum physics and specifically to quantum mechanics, where they serve as spaces of pure quantum states. Here the inner product encodes the probability amplitudes for one pure state to “collaps” to another one under measurement. When the space of pure states is of finite dimension (as is the case of interest in quantum information theory/quantum computation) then the completeness condition on a Hilbert space is automatic (see Rem. below), otherwise it naturally encodes the possibility of an infinite number of measurement outcomes.

Hilbert spaces with (bounded) linear maps between them form a (dagger-)category, often denoted Hilb or similar, with the dagger-structure given by sending bounded linear maps to their adjoint operators with respect to the Hermitian inner product. Finite-dimensional Hilbert spaces form a dagger-compact category.

See also:

• an elementary treatment of Hilbert spaces.

Definition

We first state the

• standard formulation

of the definition of Hilbert spaces and then various equivalent

• alternative formulations

Standard formulation

We first state the definition as such and then recall the ingredients that go into it:

Definition

(Hilbert space)

A Hilbert space $\big(\mathscr{H}, \langle -,-\rangle\big)$ is

1. a (complex or real) vector space $\mathscr{H}$ (possibly infinite-dimensional),

2. equipped with a Hermitian inner product $\langle -,- \rangle$ (Def. )

which is

1. complete (Def. );

2. positive definite (Def. ).

Remark

(morphisms of Hilbert spaces)
There are different notions of morphisms between Hilbert spaces:

At the very least, a morphism of Hilbert spaces is a linear map between the underlying vector spaces, and this is the default notion usually considered. Notice that such morphisms do not need to respect the inner product-structure. Often one restricts to bounded linear maps or sometimes short maps (see at maps of Banach spaces) which respect part of the inner product structure.

Instead of being fully respected by maps, the inner product on Hilbert spaces induces the dagger involution on their hom-spaces given by sending any bounded linear map $A \colon \mathscr{H}_1 \to \mathscr{H}_2$ to its Hermitian adjoint operator $A^\dagger \colon \mathscr{H}_2 \to \mathscr{H}_1$, characterized by

(1)$$\langle A^\dagger(-), -\rangle_{1} \;=\; \langle -, A(-)\rangle_{2}$$

Remark

(finite-dimensional Hilbert space)
In the case that the underlying vector space $\mathscr{H}$ happens to be finite dimensional vector space, any hermitian inner product is necessarily complete. Therefore:

A finite-dimensional Hilbert space is equivalently a positive-definite hermitian inner product space of finite dimension.

Finite-dimensional Hilbert spaces form a dagger-compact category and play a central role in quantum information theory and quantum computation, see also at finite quantum mechanics in terms of dagger-compact categories.

We recall now the meaning of the concepts entering Def. . In the following, let $\mathcal{H}$ be a vector space over the ground field $\mathbb{C}$ of complex numbers. For $z \in \mathbb{C}$ a complex number, we write $\overline{z}$ for its complex conjugate.

Remark

(alternative ground fields)
The following applies verbatim also for the ground field of real numbers $\mathbb{R}$, in which case the sesquilinear inner product below becomes bilinear and one speaks of real Hilbert spaces. Under mild assumptions, $\mathbb{C}$ and $\mathbb{R}$ are the only possible ground fields for Hilbert spaces (see MO:a/4184099).

Definition

(Hermitian inner product)
A Hermitian inner product on $\mathscr{H}$ is a function

$$\langle {-},{-} \rangle \;\colon\; \mathscr{H} \times \mathscr{H} \to \mathbb{C}$$

that is

1. sesquilinear:

   1. $\langle 0, x \rangle = 0$ and $\langle x, 0 \rangle = 0$;

   2. $\langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle$ and $\langle x, y + z \rangle = \langle x, y \rangle + \langle x, z \rangle$;

   3. $\langle c x, y \rangle = \bar{c} \langle x, y \rangle$ and $\langle x, c y \rangle = c \langle x, y \rangle$;

2. conjugate-symmetric:

   (2)$$\langle x, y \rangle = \overline{\langle y, x \rangle}$$

Remark

(convention for the inner product)
Def. uses the physicist's convention that the inner product is conjugate-linear in the first variable rather than in the second, instead of the mathematician's convention, which is the reverse. The physicist's convention fits in a little better with $2$-Hilbert spaces.

Remark

The axiom list in Def. is rather redundant. First of all, (1.1) follows from (1.3) by setting $c = 0$; besides that, (1.1–1.3) come in pairs, only one of which is needed, since each half follows from the other using (2). It is even possible to derive (1.3) from (1.2) by supposing that $V$ is a topological vector space and that the inner product is continuous (which, as we will see, is always true anyway for a Hilbert space).

Definition

(definite inner product)
Given an inner product according to Def. , consider the (norm square) function

(3)$$\array{ \mathllap{ \|{-}\|^2 \;\colon\; } \mathscr{H} &\longrightarrow& \mathbb{R} \hookrightarrow \mathbb{C} \\ v &\mapsto& \langle v, v \rangle }$$

Notice that this takes only real values, by (2).

The Hermitian inner product is called:

• positive semidefinite, or simply positive, if $\|v\|^2 \geq 0$ for all $v \in \mathscr{H}$;

• non-degenerate if $\|v\|^2 = 0$ implies that $v = 0$;

• positive definite if it is both positive and non-degenerate.

Remark

An inner product is called indefinite if some $\|v\|^2$ are positive and some are negative.

Definition

(complete inner product)
The inner product (Def. ) is complete if, given any infinite sequence $(v_1, v_2, \ldots)$ in $\mathscr{V}$ such that we have the limit

(4)$$\lim_{m,n\to\infty} \left\|\sum_{i=m}^{m+n} v_i\right\|^2 = 0 \,,$$

then there exists a (necessarily unique) sum $S$ such that

(5)$$\lim_{n\to\infty} \left\|S - \sum_{i=1}^n v_i\right\|^2 = 0 \,.$$

If the inner product is definite (Def. ), then this sum, if it exists, must be unique, and we write

$$S = \sum_{i=1}^\infty v_i$$

(with the right-hand side undefined if no such sum exists).

Alternative formulations

As Banach spaces and metric spaces

If an inner product (Def. ) is positive (Def. ), then there exists the principal square root of the norm square $\|v\|^2 = \langle v, v \rangle$ (3) being the norm $\|v\|$ of $v \in \mathscr{H}$.

This norm satisfies all of the requirements of a Banach space. It additionally satisfies the parallelogram law

(6)$$\|v_1 + v_2\|^2 + \|v_1 - v_2\|^2 \;=\; 2 \|v_1\|^2 + 2 \|v_2\|^2 \,.$$

which not all Banach spaces need satisfy. (The name of this law comes from its geometric interpretation: the norms in the left-hand side are the lengths of the diagonals of a parallelogram, while the norms in the right-hand side are the lengths of the sides.)

Furthermore, any Banach space satsifying the parallelogram law (6) has a unique inner product that reproduces the norm, defined by

$$\langle v_1, v_2 \rangle \;\coloneqq\; \frac{1}{4} \left( \|v_1 + v_2\|^2 - \|v_1 - v_2\|^2 - \mathrm{i} \|v_1 + \mathrm{i}v_2\|^2 + \mathrm{i} \|v_1 - \mathrm{i}v_2\|^2 \right) ,$$

or $\frac{1}{2}(\|v_1 + v_2\|^2 - \|v_1 - v_2\|^2)$ in the real case.

Therefore: A Hilbert space is equivalently Banach space that satisfies the parallelogram law (6).

This actually works a bit more generally; a positive semidefinite inner product space is a pseudonormed vector space that satisfies the parallelogram law. (We cannot, however, recover an indefinite inner product from a norm.)

Moreover, in any positive semidefinite inner product space, let the distance $d(x,y)$ be

$$d(x,y) \;\coloneqq\; \|y - x\| \,.$$

Then $d$ is a pseudometric; and it is a complete metric if and only if we have a Hilbert space.

In fact, the axioms of a Banach space (or pseudonormed vector space) can be written entirely in terms of the metric; we can also state the parallelogram law as follows:

$$d(x,y)^2 + d(x,-y)^2 = 2 d(x,0)^2 + 2 d(x,x+y)^2 .$$

In definitions, it is probably most common to see the metric introduced only to state the completeness requirement. Indeed, (4) says that the sequence of partial sums is a Cauchy sequence, while (5) says that the sequence of partial sums converges to $S$.

Examples

Banach spaces

All of the $p$-parametrised examples at Banach space apply if you take $p = 2$.

In particular, the $n$-dimensional vector space $\mathbb{C}^n$ is a complex Hilbert space with

$$\langle x, y \rangle = \sum_{u=1}^n \bar{x}_u y_u .$$

Any subfield $K$ of $\mathbb{C}$ gives a positive definite inner product space $K^n$ whose completion is either $\mathbb{R}^n$ or $\mathbb{C}^n$. In particular, the cartesian space $\mathbb{R}^n$ is a real Hilbert space; the geometric notions of distance and angle defined above agree with ordinary Euclidean geometry for this example.

Of Lebesgue square-integrable functions over a manifold

The L- Hilbert spaces $L^2(\mathbb{R})$, $L^2([0,1])$, $L^2(\mathbb{R}^3)$, etc (real or complex) are very well known. In general, $L^2(X)$ for $X$ a measure space consists of the almost-everywhere defined functions $f$ from $X$ to the scalar field ($\mathbb{R}$ or $\mathbb{C}$) such that $\int |f|^2$ converges to a finite number, with functions identified if they are equal almost everywhere; we have $\langle f, g\rangle = \int \bar{f} g$, which converges by the Cauchy–Schwarz inequality. In the specific cases listed (and in general, when $X$ is a locally compact Hausdorff space), we can also get this space by completing the positive definite inner product space of compactly supported continuous functions.

Of square-integrable half-densities

• canonical Hilbert space of half-densities

Properties

Bases

A basic result is that abstractly, Hilbert spaces of the same dimension are all of the same type: every Hilbert space $H$ admits an orthonormal basis, meaning a subset $S \subseteq H$ whose inclusion map extends (necessarily uniquely) to an isomorphism

$$l^2(S) \to H$$

of Hilbert spaces. Here $l^2(S)$ is the vector space consisting of those functions $x$ from $S$ to the scalar field such that

$$\|x\|^2 = \sum_{u: S} |x_u|^2$$

converges to a finite number; this may also be obtained by completing the vector space of formal linear combinations of elements of $S$ with an inner product uniquely determined by the rule

$$\langle u, v \rangle = \delta_{u v} \qquad u, v \in S$$

in which $\delta_{u v}$ denotes Kronecker delta. We thus have, in $l^2(S)$,

$$\langle x, y \rangle = \sum_{u: S} \bar{x}_u y_u .$$

(This sum converges by the Cauchy–Schwarz inequality.)

In general, this result uses the axiom of choice (usually in the form of Zorn's lemma and excluded middle) in its proof, and is equivalent to it. However, the result for separable Hilbert spaces needs only dependent choice and so is constructive by most schools' standards. Even without dependent choice, explicit orthornormal bases for particular $L^2(X)$ can often be produced using approximation of the identity techniques, often in concert with a Gram-Schmidt process.

In particular, all infinite-dimensional separable Hilbert spaces are abstractly isomorphic to $l^2(\mathbb{N})$.

Cauchy–Schwarz inequality

The Schwarz inequality (or Cauchy–Буняковский–Schwarz inequality, etc) is very handy:

$$|\langle x, y \rangle| \leq \|x\| \|y\| .$$

This is really two theorems (at least): an abstract theorem that the inequality holds in any Hilbert space, and concrete theorems that it holds when the inner product and norm are defined by the formulas used in the examples $L^2(X)$ and $l^2(S)$ above. The concrete theorems apply even to functions that don't belong to the Hilbert space and so prove that the inner product converges whenever the norms converge. (A somewhat stronger result is needed to conclude this convergence constructively; it may be found in Errett Bishop's book.)

Related concepts

• rigged Hilbert space

• Hilbert C-star-module, Hilbert bimodule

• measurable field of Hilbert spaces

• Kähler vector space

References

Hilbert spaces were effectively introduced and used by David Hilbert and others in the context of integration theory, but the terminology and the formal definition is due to:

• John von Neumann, §I in: Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Math. Ann. 102 (1930) 49–131 [doi:10.1007/BF01782338]

motivated from laying foundations for quantum mechanics:

• John von Neumann:

  Mathematische Grundlagen der Quantenmechanik, Springer (1932, 1971) [doi:10.1007/978-3-642-96048-2]

  Mathematical Foundations of Quantum Mechanics Princeton University Press (1955) [doi:10.1515/9781400889921, Wikipedia entry]

but see:

• Miklos Rédei, Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead), Studies in History and Philosophy of Modern Physics 27 4 (1996) 493-510 [doi:10.1016/S1355-2198(96)00017-2]

Early history:

• Saunders Mac Lane: Hilbert space, §5 in: Concepts and Categories, in: P. Duren, A century of mathematics in America Part 1, AMS (1988) 323-365. [pdf, ISBN:0-8218-0124-4]

Review:

• George Mackey, The Mathematical Foundations of Quamtum Mechanics A Lecture-note Volume, The mathematical physics monograph series. Princeton university (1963)

• E. Prugoveĉki, Quantum mechanics in Hilbert Space. Academic Press (1971)

An axiomatic characterization of the dagger-category Hilb of Hilbert spaces, with linear maps between them:

• Chris Heunen, Andre Kornell, Axioms for the category of Hilbert spaces, PNAS 119 9 (2022) e2117024119 [arXiv:2109.07418, doi:10.1073/pnas.2117024119]

• Chris Heunen, Andre Kornell, Nesta van der Schaaf, Axioms for the category of Hilbert spaces and linear contractions [arXiv:2211.02688]

category: analysis

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1. Dr. von Neumann, I would like to know what is a Hilbert space? – Question asked by David Hilbert, in a 1929 talk by John von Neumann in Göttingen (cf. von Neumann 1930, §I). The anecdote is narrated for instance in MacLane 1988, §5. (We have corrected ‘dann’ in the original quotation to the more likely ‘denn’, in either case expressing a certain sense of puzzlement that is not quite captured by the direct English translation.) ↩