Hilbert space

 ( Functional Analysis ) 

In mathematics, a Hilbert space is a real number or complex number inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The inner product allows lengths and angles to be defined. Furthermore, completeness means that there are enough limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space.

Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, Sequence space, consisting of generalized functions, and of holomorphic functions.

Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a linear subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to an orthonormal basis, in analogy with Cartesian coordinates in classical geometry. When this basis is countably infinite, it allows identifying the Hilbert space with the space of the infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space.

The dot product satisfies the properties

1. It is symmetric in and : .
2. It is linear function in its first argument: for any scalars , , and vectors , , and .
3. It is positive definite: for all vectors , , with equality if and only if .

An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space.However, some sources call finite-dimensional spaces with these properties pre-Hilbert spaces, reserving the term "Hilbert space" for infinite-dimensional spaces; see, e.g., . The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted , and to the angle between two vectors and by means of the formula $$\backslash mathbf\{x\}\backslash cdot\backslash mathbf\{y\}\; =\; \backslash left\backslash |\backslash mathbf\{x\}\backslash right\backslash |\; \backslash left\backslash |\backslash mathbf\{y\}\backslash right\backslash |\; \backslash ,\; \backslash cos\backslash theta\; \backslash ,.$$

Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist. A mathematical series $$\backslash sum\_\{n=0\}^\backslash infty\; \backslash mathbf\{x\}\_n$$ consisting of vectors in is absolutely convergent provided that the sum of the lengths converges as an ordinary series of real numbers:

Just as with a series of scalars, a series of vectors that converges absolutely also converges to some limit vector in the Euclidean space, in the sense that $$\backslash Biggl\backslash |\; \backslash mathbf\{L\}\; -\; \backslash sum\_\{k=0\}^N\; \backslash mathbf\{x\}\_k\; \backslash Biggr\backslash |\; \backslash to\; 0\; \backslash quad\; \backslash text\{as\; \}\; N\; \backslash to\backslash infty\; \backslash ,.$$

This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense.

Hilbert spaces are often taken over the . The complex plane denoted by is equipped with a notion of magnitude, the absolute value , which is defined as the square root of the product of with its complex conjugate: $$|z|^2\; =\; z\backslash overline\{z\}\; \backslash ,.$$

If is a decomposition of into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: $$|z|\; =\; \backslash sqrt\{x^2\; +\; y^2\}\; \backslash ,.$$

The inner product of a pair of complex numbers and is the product of with the complex conjugate of : $$\backslash langle\; z,\; w\backslash rangle\; =\; z\backslash overline\{w\}\backslash ,.$$

This is complex-valued. The real part of gives the usual two-dimensional Euclidean dot product.

A second example is the space whose elements are pairs of complex numbers . Then an inner product of with another such vector is given by $$\backslash langle\; z,\; w\backslash rangle\; =\; z\_1\backslash overline\{w\_1\}\; +\; z\_2\backslash overline\{w\_2\}\backslash ,.$$

The real part of is then the four-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging and is the complex conjugate: $$\backslash langle\; w,\; z\backslash rangle\; =\; \backslash overline\{\backslash langle\; z,\; w\backslash rangle\}\backslash ,.$$

To say that a complex vector space is a means that there is an inner product $\backslash langle\; x,\; y\; \backslash rangle$ associating a complex number to each pair of elements $x,\; y$ of that satisfies the following properties:

1. The inner product is conjugate symmetric; that is, the inner product of a pair of elements is equal to the complex conjugate of the inner product of the swapped elements: $$\backslash langle\; y,\; x\backslash rangle\; =\; \backslash overline\{\backslash langle\; x,\; y\backslash rangle\}\backslash ,.$$ Importantly, this implies that $\backslash langle\; x,\; x\backslash rangle$ is a real number.
2. The inner product is linear in its firstIn some conventions, inner products are linear in their second arguments instead. argument. For all complex numbers $a$ and $b,$ $$\backslash langle\; ax\_1\; +\; bx\_2,\; y\backslash rangle\; =\; a\backslash langle\; x\_1,\; y\backslash rangle\; +\; b\backslash langle\; x\_2,\; y\backslash rangle\backslash ,.$$
3. The inner product of an element with itself is positive definite: $$\backslash begin\{alignat\}\{4\}$$

 \langle x, x\rangle > 0 & \quad \text{ if } x \neq 0, \\
 \langle x, x\rangle = 0 & \quad \text{ if } x = 0\,.
     

It follows from properties 1 and 2 that a complex inner product is , also called , in its second argument, meaning that $$\backslash langle\; x,\; ay\_1\; +\; by\_2\backslash rangle\; =\; \backslash bar\{a\}\backslash langle\; x,\; y\_1\backslash rangle\; +\; \backslash bar\{b\}\backslash langle\; x,\; y\_2\backslash rangle\backslash ,.$$

A is defined in the same way, except that is a real vector space and the inner product takes real values. Such an inner product will be a bilinear map and $(H,\; H,\; \backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle)$ will form a dual system.

The norm is the real-valued function $$\backslash |x\backslash |\; =\; \backslash sqrt\{\backslash langle\; x,\; x\; \backslash rangle\}\backslash ,,$$ and the distance $d$ between two points $x,\; y$ in is defined in terms of the norm by $$d(x,\; y)\; =\; \backslash |x\; -\; y\backslash |\; =\; \backslash sqrt\{\backslash langle\; x\; -\; y,\; x\; -\; y\; \backslash rangle\}\backslash ,.$$

That this function is a distance function means firstly that it is symmetric in $x$ and $y,$ secondly that the distance between $x$ and itself is zero, and otherwise the distance between $x$ and $y$ must be positive, and lastly that the triangle inequality holds, meaning that the length of one leg of a triangle cannot exceed the sum of the lengths of the other two legs: $$d(x,\; z)\; \backslash leq\; d(x,\; y)\; +\; d(y,\; z)\backslash ,.$$

This last property is ultimately a consequence of the more fundamental Cauchy–Schwarz inequality, which asserts $$\backslash left|\backslash langle\; x,\; y\backslash rangle\backslash right|\; \backslash leq\; \backslash |x\backslash |\; \backslash |y\backslash |$$ with equality if and only if $x$ and $y$ are linearly dependent.

With a distance function defined in this way, any inner product space is a metric space, and sometimes is known as a . Any pre-Hilbert space that is additionally also a complete space is a Hilbert space.

The of is expressed using a form of the Cauchy criterion for sequences in : a pre-Hilbert space is complete if every Cauchy sequence converges with respect to this norm to an element in the space. Completeness can be characterized by the following equivalent condition: if a series of vectors $$\backslash sum\_\{k=0\}^\backslash infty\; u\_k$$ converges absolutely in the sense that then the series converges in , in the sense that the partial sums converge to an element of .

As a complete normed space, Hilbert spaces are by definition also . As such they are topological vector spaces, in which topology notions like the open set and closed set of subsets are well defined. Of special importance is the notion of a closed linear subspace of a Hilbert space that, with the inner product induced by restriction, is also complete (being a closed set in a complete metric space) and therefore a Hilbert space in its own right.

The inner product on is defined by: $$\backslash langle\; \backslash mathbf\{z\},\; \backslash mathbf\{w\}\backslash rangle\; =\; \backslash sum\_\{n=1\}^\backslash infty\; z\_n\backslash overline\{w\_n\}\backslash ,,$$

This second series converges as a consequence of the Cauchy–Schwarz inequality and the convergence of the previous series.

Completeness of the space holds provided that whenever a series of elements from converges absolutely (in norm), then it converges to an element of . The proof is basic in mathematical analysis, and permits mathematical series of elements of the space to be manipulated with the same ease as series of complex numbers (or vectors in a finite-dimensional Euclidean space).

In the first decade of the 20th century, parallel developments led to the introduction of Hilbert spaces. The first of these was the observation, which arose during David Hilbert and Erhard Schmidt's study of integral equations, that two square-integrable real-valued functions and on an interval have an inner product

$\backslash langle\; f,\; g\; \backslash rangle\; =\; \backslash int\_a^b\; f(x)g(x)\backslash ,\; \backslash mathrm\{d\}x$

$f(x)\; \backslash mapsto\; \backslash int\_a^b\; K(x,\; y)\; f(y)\backslash ,\; \backslash mathrm\{d\}y$

$K(x,\; y)\; =\; \backslash sum\_n\; \backslash lambda\_n\backslash varphi\_n(x)\backslash varphi\_n(y)$

The second development was the Lebesgue integral, an alternative to the Riemann integral introduced by Henri Lebesgue in 1904.. Further details on the history of integration theory can be found in and . The Lebesgue integral made it possible to integrate a much broader class of functions. In 1907, Frigyes Riesz and Ernst Sigismund Fischer independently proved that the space of square Lebesgue-integrable functions is a complete metric space. As a consequence of the interplay between geometry and completeness, the 19th century results of Joseph Fourier, Friedrich Bessel and Marc-Antoine Parseval on trigonometric series easily carried over to these more general spaces, resulting in a geometrical and analytical apparatus now usually known as the Riesz–Fischer theorem.

Further basic results were proved in the early 20th century. For example, the Riesz representation theorem was independently established by Maurice Fréchet and Frigyes Riesz in 1907.In , the result that every linear functional on is represented by integration is jointly attributed to and . The general result, that the dual of a Hilbert space is identified with the Hilbert space itself, can be found in . John von Neumann coined the term abstract Hilbert space in his work on unbounded Hermitian operators. Although other mathematicians such as Hermann Weyl and Norbert Wiener had already studied particular Hilbert spaces in great detail, often from a physically motivated point of view, von Neumann gave the first complete and axiomatic treatment of them. Von Neumann later used them in his seminal work on the foundations of quantum mechanics, and in his continued work with Eugene Wigner. The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups.

The significance of the concept of a Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of quantum mechanics. In short, the states of a quantum mechanical system are vectors in a certain Hilbert space, the observables are hermitian operators on that space, the symmetry of the system are , and measurements are orthogonal projections. The relation between quantum mechanical symmetries and unitary operators provided an impetus for the development of the unitary representation theory of groups, initiated in the 1928 work of Hermann Weyl. On the other hand, in the early 1930s it became clear that classical mechanics can be described in terms of Hilbert space (Koopman–von Neumann classical mechanics) and that certain properties of classical dynamical systems can be analyzed using Hilbert space techniques in the framework of ergodic theory.

The algebra of in quantum mechanics is naturally an algebra of operators defined on a Hilbert space, according to Werner Heisenberg's matrix mechanics formulation of quantum theory. Von Neumann began investigating in the 1930s, as rings of operators on a Hilbert space. The kind of algebras studied by von Neumann and his contemporaries are now known as von Neumann algebras. In the 1940s, Israel Gelfand, Mark Naimark and Irving Segal gave a definition of a kind of operator algebras called C*-algebras that on the one hand made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the operator algebras that had previously been studied. The spectral theorem for self-adjoint operators in particular that underlies much of the existing Hilbert space theory was generalized to C*-algebras. These techniques are now basic in abstract harmonic analysis and representation theory.

The inner product of functions and in is then defined as $$\backslash langle\; f,\; g\backslash rangle\; =\; \backslash int\_X\; f(t)\; \backslash overline\{g(t)\}\; \backslash ,\; \backslash mathrm\{d\}\; \backslash mu(t)$$ or $$\backslash langle\; f,\; g\backslash rangle\; =\; \backslash int\_X\; \backslash overline\{f(t)\}\; g(t)\; \backslash ,\; \backslash mathrm\{d\}\; \backslash mu(t)\; \backslash ,,$$

where the second form (conjugation of the first element) is commonly found in the theoretical physics literature. For and in , the integral exists because of the Cauchy–Schwarz inequality, and defines an inner product on the space. Equipped with this inner product, is in fact complete. The Lebesgue integral is essential to ensure completeness: on domains of real numbers, for instance, not enough functions are Riemann integral.

The Lebesgue spaces appear in many natural settings. The spaces and of square-integrable functions with respect to the Lebesgue measure on the real line and unit interval, respectively, are natural domains on which to define the Fourier transform and Fourier series. In other situations, the measure may be something other than the ordinary Lebesgue measure on the real line. For instance, if is any positive measurable function, the space of all measurable functions on the interval satisfying is called the weighted space , and is called the weight function. The inner product is defined by $$\backslash langle\; f,\; g\backslash rangle\; =\; \backslash int\_0^1\; f(t)\; \backslash overline\{g(t)\}\; w(t)\; \backslash ,\; \backslash mathrm\{d\}t\; \backslash ,.$$

The weighted space is identical with the Hilbert space where the measure of a Lebesgue-measurable set is defined by $$\backslash mu(A)\; =\; \backslash int\_A\; w(t)\backslash ,\backslash mathrm\{d\}t\; \backslash ,.$$

Weighted spaces like this are frequently used to study orthogonal polynomials, because different families of orthogonal polynomials are orthogonal with respect to different weighting functions.

 \alpha y(a)+\alpha' y'(a) &= 0 \\
 \beta y(b) + \beta' y'(b) &= 0 \,.
     

In particular, when is not equal to , one can find a nonzero vector orthogonal to (select and ). A very useful criterion is obtained by applying this observation to the closed subspace generated by a subset of .

A subset of spans a dense vector subspace if (and only if) the vector 0 is the sole vector orthogonal to .

The Riesz representation theorem affords a convenient description of the dual space. To every element of , there is a unique element of , defined by $$\backslash varphi\_u(x)\; =\; \backslash langle\; x,\; u\backslash rangle$$ where moreover, $\backslash left\backslash |\; \backslash varphi\_u\; \backslash right\backslash |\; =\; \backslash left\backslash |\; u\; \backslash right\backslash |.$

The Riesz representation theorem states that the map from to defined by is Surjective map, which makes this map an Isometry Antilinear map isomorphism. So to every element of the dual there exists one and only one in such that $$\backslash langle\; x,\; u\_\backslash varphi\backslash rangle\; =\; \backslash varphi(x)$$ for all . The inner product on the dual space satisfies $$\backslash langle\; \backslash varphi,\; \backslash psi\; \backslash rangle\; =\; \backslash langle\; u\_\backslash psi,\; u\_\backslash varphi\; \backslash rangle\; \backslash ,.$$

The reversal of order on the right-hand side restores linearity in from the antilinearity of . In the real case, the antilinear isomorphism from to its dual is actually an isomorphism, and so real Hilbert spaces are naturally isomorphic to their own duals.

The representing vector is obtained in the following way. When , the kernel is a closed vector subspace of , not equal to , hence there exists a nonzero vector orthogonal to . The vector is a suitable scalar multiple of . The requirement that yields $$u\; =\; \backslash langle\; v,\; v\; \backslash rangle^\{-1\}\; \backslash ,\; \backslash overline\{\backslash varphi\; (v)\}\; \backslash ,\; v\; \backslash ,.$$

This correspondence is exploited by the bra–ket notation popular in physics. It is common in physics to assume that the inner product, denoted by , is linear on the right, $$\backslash langle\; x\; |\; y\; \backslash rangle\; =\; \backslash langle\; y,\; x\; \backslash rangle\; \backslash ,.$$ The result can be seen as the action of the linear functional (the bra) on the vector (the ket).

The Riesz representation theorem relies fundamentally not just on the presence of an inner product, but also on the completeness of the space. In fact, the theorem implies that the Banach space of any inner product space can be identified with its completion. An immediate consequence of the Riesz representation theorem is also that a Hilbert space is reflexive space, meaning that the natural map from into its dual space is an isomorphism.

For example, any orthonormal sequence converges weakly to 0, as a consequence of Bessel's inequality. Every weakly convergent sequence is bounded, by the uniform boundedness principle.

Conversely, every bounded sequence in a Hilbert space admits weakly convergent subsequences (Alaoglu's theorem). This fact may be used to prove minimization results for continuous , in the same way that the Bolzano–Weierstrass theorem is used for continuous functions on . Among several variants, one simple statement is as follows:

If is a convex continuous function such that tends to when tends to , then admits a minimum at some point .

This fact (and its various generalizations) are fundamental for direct methods in the calculus of variations. Minimization results for convex functionals are also a direct consequence of the slightly more abstract fact that closed bounded convex subsets in a Hilbert space are Weak topology, since is reflexive. The existence of weakly convergent subsequences is a special case of the Eberlein–Šmulian theorem.

The (geometrical) Hahn–Banach theorem asserts that a closed convex set can be separated from any point outside it by means of a hyperplane of the Hilbert space. This is an immediate consequence of the best approximation property: if is the element of a closed convex set closest to , then the separating hyperplane is the plane perpendicular to the segment passing through its midpoint.

The sum and the composite of two bounded linear operators is again bounded and linear. For y in H₂, the map that sends to is linear and continuous, and according to the Riesz representation theorem can therefore be represented in the form $$\backslash left\backslash langle\; x,\; A^*\; y\; \backslash right\backslash rangle\; =\; \backslash langle\; Ax,\; y\; \backslash rangle$$ for some vector in . This defines another bounded linear operator , the adjoint of . The adjoint satisfies . When the Riesz representation theorem is used to identify each Hilbert space with its continuous dual space, the adjoint of can be shown to be identical to the transpose of , which by definition sends $\backslash psi\; \backslash in\; H\_2^\{*\}$ to the functional $\backslash psi\; \backslash circ\; A\; \backslash in\; H\_1^\{*\}.$

The set of all bounded linear operators on (meaning operators ), together with the addition and composition operations, the norm and the adjoint operation, is a C*-algebra, which is a type of operator algebra.

An element of is called 'self-adjoint' or 'Hermitian' if . If is Hermitian and for every , then is called 'nonnegative', written ; if equality holds only when , then is called 'positive'. The set of self adjoint operators admits a partial order, in which if . If has the form for some , then is nonnegative; if is invertible, then is positive. A converse is also true in the sense that, for a non-negative operator , there exists a unique non-negative square root such that $$A\; =\; B^2\; =\; B^*B\backslash ,.$$

In a sense made precise by the spectral theorem, self-adjoint operators can usefully be thought of as operators that are "real". An element of is called normal if . Normal operators decompose into the sum of a self-adjoint operator and an imaginary multiple of a self adjoint operator $$A\; =\; \backslash frac\{A\; +\; A^*\}\{2\}\; +\; i\backslash frac\{A\; -\; A^*\}\{2i\}$$ that commute with each other. Normal operators can also usefully be thought of in terms of their real and imaginary parts.

An element of is called unitary operator if is invertible and its inverse is given by . This can also be expressed by requiring that be onto and for all . The unitary operators form a group under composition, which is the isometry group of .

An element of is compact operator if it sends bounded sets to relatively compact sets. Equivalently, a bounded operator is compact if, for any bounded sequence , the sequence has a convergent subsequence. Many integral operators are compact, and in fact define a special class of operators known as Hilbert–Schmidt operators that are especially important in the study of integral equations. Fredholm operators differ from a compact operator by a multiple of the identity, and are equivalently characterized as operators with a finite dimensional kernel and cokernel. The index of a Fredholm operator is defined by $$\backslash operatorname\{index\}\; T\; =\; \backslash dim\backslash ker\; T\; -\; \backslash dim\backslash operatorname\{coker\}\; T\; \backslash ,.$$

The index is homotopy invariant, and plays a deep role in differential geometry via the Atiyah–Singer index theorem.

The adjoint of a densely defined unbounded operator is defined in essentially the same manner as for bounded operators. Self-adjoint unbounded operators play the role of the observables in the mathematical formulation of quantum mechanics. Examples of self-adjoint unbounded operators on the Hilbert space are:

• A suitable extension of the differential operator $$(A\; f)(x)\; =\; -i\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\; f(x)\; \backslash ,,$$ where is the imaginary unit and is a differentiable function of compact support.
• The multiplication-by- operator: $$(B\; f)\; (x)\; =\; x\; f(x)\backslash ,.$$

These correspond to the momentum and position observables, respectively. Neither nor is defined on all of , since in the case of the derivative need not exist, and in the case of the product function need not be square integrable. In both cases, the set of possible arguments form dense subspaces of .

consisting of the set of all where , , and inner product defined by $$\backslash bigl\backslash langle\; (x\_1,\; x\_2),\; (y\_1,\; y\_2)\backslash bigr\backslash rangle\_\{H\_1\; \backslash oplus\; H\_2\}\; =\; \backslash left\backslash langle\; x\_1,\; y\_1\backslash right\backslash rangle\_\{H\_1\}\; +\; \backslash left\backslash langle\; x\_2,\; y\_2\backslash right\backslash rangle\_\{H\_2\}\; \backslash ,.$$

More generally, if is a family of Hilbert spaces indexed by , then the direct sum of the , denoted $$\backslash bigoplus\_\{i\; \backslash in\; I\}H\_i$$ consists of the set of all indexed families $$x\; =\; (x\_i\; \backslash in\; H\_i\; \backslash mid\; i\; \backslash in\; I)\; \backslash in\; \backslash prod\_\{i\; \backslash in\; I\}H\_i$$ in the Cartesian product of the such that

The inner product is defined by $$\backslash langle\; x,\; y\backslash rangle\; =\; \backslash sum\_\{i\; \backslash in\; I\}\; \backslash left\backslash langle\; x\_i,\; y\_i\backslash right\backslash rangle\_\{H\_i\}\; \backslash ,.$$

Each of the is included as a closed subspace in the direct sum of all of the . Moreover, the are pairwise orthogonal. Conversely, if there is a system of closed subspaces, , , in a Hilbert space , that are pairwise orthogonal and whose union is dense in , then is canonically isomorphic to the direct sum of . In this case, is called the internal direct sum of the . A direct sum (internal or external) is also equipped with a family of orthogonal projections onto the th direct summand . These projections are bounded, self-adjoint, idempotent operators that satisfy the orthogonality condition $$E\_i\; E\_j\; =\; 0,\backslash quad\; i\; \backslash neq\; j\; \backslash ,.$$

The spectral theorem for compact operator self-adjoint operators on a Hilbert space states that splits into an orthogonal direct sum of the eigenspaces of an operator, and also gives an explicit decomposition of the operator as a sum of projections onto the eigenspaces. The direct sum of Hilbert spaces also appears in quantum mechanics as the Fock space of a system containing a variable number of particles, where each Hilbert space in the direct sum corresponds to an additional degree of freedom for the quantum mechanical system. In representation theory, the Peter–Weyl theorem guarantees that any unitary representation of a compact group on a Hilbert space splits as the direct sum of finite-dimensional representations.

This formula then extends by sesquilinearity to an inner product on . The Hilbertian tensor product of and , sometimes denoted by , is the Hilbert space obtained by completing for the metric associated to this inner product.

An example is provided by the Hilbert space . The Hilbertian tensor product of two copies of is isometrically and linearly isomorphic to the space of square-integrable functions on the square . This isomorphism sends a simple tensor to the function $$(s,\; t)\; \backslash mapsto\; f\_1(s)\; \backslash ,\; f\_2(t)$$ on the square.

This example is typical in the following sense. Associated to every simple tensor product is the rank one operator from to that maps a given as $$x^*\; \backslash mapsto\; x^*(x\_1)\; x\_2\; \backslash ,.$$

This mapping defined on simple tensors extends to a linear identification between and the space of finite rank operators from to . This extends to a linear isometry of the Hilbertian tensor product with the Hilbert space of Hilbert–Schmidt operators from to .

1. Orthogonality: Every two different elements of are orthogonal: for all with .
2. Normalization: Every element of the family has norm 1: for all .
3. Completeness: The linear span of the family , , is dense set in H.

A system of vectors satisfying the first two conditions basis is called an orthonormal system or an orthonormal set (or an orthonormal sequence if is countable set). Such a system is always linearly independent.

Despite the name, an orthonormal basis is not, in general, a basis in the sense of linear algebra (Hamel basis). More precisely, an orthonormal basis is a Hamel basis if and only if the Hilbert space is a finite-dimensional vector space.

Completeness of an orthonormal system of vectors of a Hilbert space can be equivalently restated as:

for every , if for all , then .

This is related to the fact that the only vector orthogonal to a dense linear subspace is the zero vector, for if is any orthonormal set and is orthogonal to , then is orthogonal to the closure of the linear span of , which is the whole space.

Examples of orthonormal bases include:

• the set forms an orthonormal basis of with the dot product;
• the sequence with forms an orthonormal basis of the complex space ;

In the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish the two, the latter basis is also called a Hamel basis. That the span of the basis vectors is dense implies that every vector in the space can be written as the sum of an infinite series, and the orthogonality implies that this decomposition is unique.

 e_1 &= (1, 0, 0, \dots) \\
 e_2 &= (0, 1, 0, \dots) \\
     & \ \ \vdots
     

 1 & \text{if } b=b'\\
 0 & \text{otherwise.}