The Hilbert Space L2 and the Hilbert Cube

Definition and illustration of Hilbert Space

Motivating example: Euclidean vector space

One of the most familiar examples of a Hilbert space is the Euclidean vector space consisting of three-dimensional vectors, denoted by R³, and equipped with the dot product. The dot product takes two vectors x and y, and produces a real number x ⋅ y. If x and y are represented in Cartesian coordinates, then the dot product is defined by

${\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\,.$

The dot product satisfies the properties:

1. It is symmetric in x and y: x ⋅ y = y ⋅ x.
2. It is linear in its first argument: (ax₁ + bx₂) ⋅ y = ax₁ ⋅ y + bx₂ ⋅ y for any scalars a, b, and vectors x₁, x₂, and y.
3. It is positive definite: for all vectors x, x ⋅ x ≥ 0 , with equality if and only if x = 0.

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. 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 $||{\textbf {x}}||$, and to the angle θ between two vectors x and y by means of the formula

$\mathbf {x} \cdot \mathbf {y} =\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|\,\cos \theta \,.$

Completeness means that if a particle moves along the broken path (in blue) travelling a finite total distance, then the particle has a well-defined net displacement (in orange).

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

$\sum _{n=0}^{\infty }\mathbf {x} _{n}$

consisting of vectors in R³ is absolutely convergent provided that the sum of the lengths converges as an ordinary series of real numbers:

$\sum _{k=0}^{\infty }\|\mathbf {x} _{k}\|<\infty \,.$

Just as with a series of scalars, a series of vectors that converges absolutely also converges to some limit vector L in the Euclidean space, in the sense that

$\left\|\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}\right\|\to 0\quad {\text{as }}N\to \infty \,.$

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 complex numbers. The complex plane denoted by C is equipped with a notion of magnitude, the complex modulus $|z|$ which is defined as the square root of the product of z with its complex conjugate:

$|z|^{2}=z{\overline {z}}\,.$

If z = x + iy is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length:

$|z|={\sqrt {x^{2}+y^{2}}}\,.$

The inner product of a pair of complex numbers z and w is the product of z with the complex conjugate of w:

$\langle z,w\rangle =z{\overline {w}}\,.$

This is complex-valued. The real part of ⟨z, w⟩ gives the usual two-dimensional Euclidean dot product.

A second example is the space C² whose elements are pairs of complex numbers z = (z₁, z₂). Then the inner product of z with another such vector w = (w₁, w₂) is given by

$\langle z,w\rangle =z_{1}{\overline {w_{1}}}+z_{2}{\overline {w_{2}}}\,.$

The real part of ⟨z, w⟩ is then the two-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate:

$\langle w,z\rangle ={\overline {\langle z,w\rangle }}\,.$

Definition

A Hilbert space H is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product.

To say that H is a complex inner product space means that H is a complex vector space on which there is an inner product ⟨x, y⟩ associating a complex number to each pair of elements x, y of H 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:
   $$\langle y,x\rangle ={\overline {\langle x,y\rangle }}\,.$$
2. The inner product is linear in its first argument. For all complex numbers a and b,
   $$\langle ax_{1}+bx_{2},y\rangle =a\langle x_{1},y\rangle +b\langle x_{2},y\rangle \,.$$
3. The inner product of an element with itself is positive definite:
   $${\begin{cases}\langle x,x\rangle >0&x\neq 0\\\langle x,x\rangle =0&x=0\,.\end{cases}}$$
   (Note that property 1 implies that $\langle x,x\rangle$ is real.)

It follows from properties 1 and 2 that a complex inner product is antilinear, also called conjugate linear, in its second argument, meaning that

$\langle x,ay_{1}+by_{2}\rangle ={\bar {a}}\langle x,y_{1}\rangle +{\bar {b}}\langle x,y_{2}\rangle \,.$

A real inner product space is defined in the same way, except that H is a real vector space and the inner product takes real values. Such an inner product will be a bilinear map and (H, H, ⟨⋅, ⋅⟩) will form a dual system.

The norm is the real-valued function

$\|x\|={\sqrt {\langle x,x\rangle }}\,,$

and the distance d between two points x, y in H is defined in terms of the norm by

$d(x,y)=\|x-y\|={\sqrt {\langle x-y,x-y\rangle }}\,.$

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 xyz cannot exceed the sum of the lengths of the other two legs:

$d(x,z)\leq d(x,y)+d(y,z)\,.$

This last property is ultimately a consequence of the more fundamental Cauchy–Schwarz inequality, which asserts

${\bigl |}\langle x,y\rangle {\bigr |}\leq \left\|x\right\|\left\|y\right\|$

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 Hausdorff pre-Hilbert space. Any pre-Hilbert space that is additionally also a complete space is a Hilbert space.

The completeness of H is expressed using a form of the Cauchy criterion for sequences in H: a pre-Hilbert space H 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

$\sum _{k=0}^{\infty }u_{k}$

converges absolutely in the sense that

$\sum _{k=0}^{\infty }\|u_{k}\|<\infty \,,$

then the series converges in H, in the sense that the partial sums converge to an element of H.

As a complete normed space, Hilbert spaces are by definition also Banach spaces. As such they are topological vector spaces, in which topological notions like the openness and closedness 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.

Second example: sequence spaces

The sequence space l² consists of all infinite sequences z = (z₁, z₂, ...) of complex numbers such that the series

$\sum _{n=1}^{\infty }|z_{n}|^{2}$

converges. The inner product on l² is defined by

$\langle \mathbf {z} ,\mathbf {w} \rangle =\sum _{n=1}^{\infty }z_{n}{\overline {w_{n}}}\,,$

with the latter series converging as a consequence of the Cauchy–Schwarz inequality.

Completeness of the space holds provided that whenever a series of elements from l² converges absolutely (in norm), then it converges to an element of l². 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).

Lebesgue spaces

Lebesgue spaces are function spaces associated to measure spaces (X, M, μ), where X is a set, M is a σ-algebra of subsets of X, and μ is a countably additive measure on M. Let L²(X, μ) be the space of those complex-valued measurable functions on X for which the Lebesgue integral of the square of the absolute value of the function is finite, i.e., for a function f in L²(X, μ),

$\int _{X}|f|^{2}\mathrm {d} \mu <\infty \,,$

and where functions are identified if and only if they differ only on a set of measure zero.

The inner product of functions f and g in L²(X, μ) is then defined as

$\langle f,g\rangle =\int _{X}f(t){\overline {g(t)}}\,\mathrm {d} \mu (t)\$ or $\ \int _{X}{\overline {f(t)}}g(t)\,\mathrm {d} \mu (t)\,,$

where the second form (conjugation of the first element) is commonly found in the theoretical physics literature. For f and g in L², the integral exists because of the Cauchy–Schwarz inequality, and defines an inner product on the space. Equipped with this inner product, L² is in fact complete. The Lebesgue integral is essential to ensure completeness: on domains of real numbers, for instance, not enough functions are Riemann integrable.

The Lebesgue spaces appear in many natural settings. The spaces L²(R) and L²([0,1]) 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 w is any positive measurable function, the space of all measurable functions f on the interval [0, 1] satisfying

$\int _{0}^{1}{\bigl |}f(t){\bigr |}^{2}w(t)\,\mathrm {d} t<\infty$

is called the weighted L² space L²_w([0, 1])}}, and w is called the weight function. The inner product is defined by

$\langle f,g\rangle =\int _{0}^{1}f(t){\overline {g(t)}}w(t)\,\mathrm {d} t\,.$

The weighted space L²_w([0, 1])}} is identical with the Hilbert space L²([0, 1], μ) where the measure μ of a Lebesgue-measurable set A is defined by

$\mu (A)=\int _{A}w(t)\,\mathrm {d} t\,.$

Weighted L² spaces like this are frequently used to study orthogonal polynomials, because different families of orthogonal polynomials are orthogonal with respect to different weighting functions.

Hilbert Cube

In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube (see below).

Definition

The Hilbert cube is best defined as the topological product of the intervals [0, 1/n] for n = 1, 2, 3, 4, ... That is, it is a cuboid of countably infinite dimension, where the lengths of the edges in each orthogonal direction form the sequence $\lbrace 1/n\rbrace _{n\in \mathbb {N} }$.

The Hilbert cube is homeomorphic to the product of countably infinitely many copies of the unit interval [0, 1]. In other words, it is topologically indistinguishable from the unit cube of countably infinite dimension.

If a point in the Hilbert cube is specified by a sequence $\lbrace a_{n}\rbrace$ with $0\leq a_{n}\leq 1/n$, then a homeomorphism to the infinite dimensional unit cube is given by $h(a)_{n}=n\cdot a_{n}$.

The Hilbert cube as a metric space

It is sometimes convenient to think of the Hilbert cube as a metric space, indeed as a specific subset of a separable Hilbert space (i.e. a Hilbert space with a countably infinite Hilbert basis). For these purposes, it is best not to think of it as a product of copies of [0,1], but instead as

[0,1] × [0,1/2] × [0,1/3] × ···;

as stated above, for topological properties, this makes no difference. That is, an element of the Hilbert cube is an infinite sequence

(x_n)

that satisfies

0 ≤ x_n ≤ 1/n.

Any such sequence belongs to the Hilbert space ℓ₂, so the Hilbert cube inherits a metric from there. One can show that the topology induced by the metric is the same as the product topology in the above definition.

Properties

As a product of compact Hausdorff spaces, the Hilbert cube is itself a compact Hausdorff space as a result of the Tychonoff theorem. The compactness of the Hilbert cube can also be proved without the Axiom of Choice by constructing a continuous function from the usual Cantor set onto the Hilbert cube.

In ℓ₂, no point has a compact neighbourhood (thus, ℓ₂ is not locally compact). One might expect that all of the compact subsets of ℓ₂ are finite-dimensional. The Hilbert cube shows that this is not the case. But the Hilbert cube fails to be a neighbourhood of any point p because its side becomes smaller and smaller in each dimension, so that an open ball around p of any fixed radius e > 0 must go outside the cube in some dimension.

Any infinite-dimensional convex compact subset of $l_{2}$ is homeomorphic to the Hilbert cube. The Hilbert cube is a convex set, whose span is the whole space, but whose interior is empty. This situation is impossible in finite dimensions. The tangent cone to the cube at the zero vector is the whole space.

Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable (and therefore T4) and second countable. It is more interesting that the converse also holds: Every second countable T4 space is homeomorphic to a subset of the Hilbert cube.

Every G_δ-subset of the Hilbert cube is a Polish space, a topological space homeomorphic to a separable and complete metric space. Conversely, every Polish space is homeomorphic to a G_δ-subset of the Hilbert cube.

Licensing

Content obtained and/or adapted from:

• Hilbert space, Wikipedia under a CC BY-SA license
• Lp space, Wikipedia under a CC BY-SA license
• Hilbert cube, Wikipedia under a CC BY-SA license