nLab
Hilbert space

Idea

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

A Hilbert space is a (possibly) infinite-dimensional generalisation of the traditional spaces of Euclidean geometry in which the notions of distance and angle still make good sense. This is done through an algebraic operation, the inner product, that generalises the dot product.

Hilbert spaces were made famous to the world at large through their applications to physics, where they organise the pure states of quantum systems.

See also

• an elementary treatment of Hilbert spaces.

Definitions

Let $V$ be a vector space over the field of complex numbers. (One can generalise the choice of field somewhat.) An inner product (in the most general, possibly indefinite, sense) on $V$ is a function

$$\langle {-},{-} \rangle: V \times V \to \mathbb{C}$$

that is (1–3) sesquilinear and (4) conjugate-symmetric; that is:

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 $\langle x, c y \rangle = c \langle x, y \rangle$;
4. $\langle x, y \rangle = \overline{\langle y, x \rangle}$.

Here we use the physicist's convention that the inner product is conjugate-linear in the first variable rather than in the second, rather than the mathematician's convention, which is the reverse. The physicist's convention fits in a little better with $2$-Hilbert spaces. Note that we use the same field as values of the inner product as for scalars; the complex conjugation will be irrelevant for some choices of field.

The axiom list above is rather redundant. First of all, (1) follows from (3) by setting $c = 0$; besides that, (1–3) come in pairs, only one of which is needed, since each half follows from the other using (4). It is even possible to derive (3) from (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).

The next concept to define is (semi)definiteness. We define a function $\|{-}\|^2: V \to \mathbb{C}$ by $\|x\|^2 = \langle x, x \rangle$; in fact, $\|{-}\|^2$ takes only real values, by (4). * The inner product is positive semidefinite, or simply positive, if $\|x\|^2 \geq 0$ always. * Notice that (by 1), $\|x\|^2 = 0$ if $x = 0$; the inner product is definite if the converse holds. * An inner product is positive definite if it is both positive and definite. * As an aside, there are also negative (semi)definite inner products, which are slightly less convenient but not really different. An inner product is indefinite if some $\|x\|^2$ are positive and some are negative; these have a very different flavour.

The inner product is complete if, given any infinite sequence $(v_1, v_2, \ldots)$ such that

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

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

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

If the inner product is definite, 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).

Then a Hilbert space is simply a vector space equipped with a complete positive definite inner product.

Hilbert spaces as Banach spaces

If an inner product is positive, then we can take the principal square root of $\|x\|^2 = \langle x, x \rangle$ to get the a real number $\|x\|$, the norm of $x$.

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

$$\|x + y\|^2 + \|x - y\|^2 = 2 \|x\|^2 + 2 \|y\|^2 ,$$

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 has a unique inner product that reproduces the norm, defined by

$$\langle x, y \rangle = \frac{1}{4}\left(\|x + y\|^2 - \|x - y\|^2 - \mathrm{i} \|x + \mathrm{i}y\|^2 + \mathrm{i} \|x - \mathrm{i}y\|^2\right) ,$$

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

Therefore, it is possible to define a Hilbert space as a Banach space that satisfies the parallelogram law. 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.)

Hilbert spaces as metric spaces

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

$$d(x,y) = \|y - x\| .$$

Then $d$ is a pseudometric; 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, (1) says that the sequence of partial sums is a Cauchy sequence, while (2) says that the sequence of partial sums converges to $S$.

Hilbert spaces as conformal spaces

Given two vectors $x$ and $y$, both nonzero, let the angle between them be the angle $\theta(x,y)$ whose cosine is

$$\cos \theta(x,y) = \frac { \langle x, y \rangle } { \|x\| \|y\| } .$$

(Note that this angle may be imaginary in general, but not for a Hilbert space over $\mathbb{R}$.)

A Hilbert space cannot be reconstructed entirely from its angles, however (even given the underlying vector space). The inner product can only be recovered up to a positive scale factor.

Morphisms of Hilbert spaces

See discussion at Banach space. There is more to be said here concerning duals (including why the theory of Hilbert spaces is slightly nicer over $\mathbb{C}$ while that of Banach spaces is slightly nicer over $\mathbb{R}$).

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 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

References

Standard accounts of Hilbert spaces in quantum mechanics include

• John von Neumann, Mathematische Grundlagen der Quantenmechanik. (German) Mathematical Foundations of Quantum Mechanics. Berlin, Germany: Springer Verlag, 1932.

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

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

category: analysis

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1. Dr. von Neumann, I would like to know what is a Hilbert space ? Question asked by Hilbert in a 1929 talk by v. Neumann in Göttingen. The anecdote is narrated together with additional information on the introduction of adjoint operators to quantum mechanics by Saunders Mac Lane in Concepts and Categories (link, p.330). Note, that we have corrected ‘dann’ in the original quotation to the more likely ‘denn’. ↩