# Hilbert space

The state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space.

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.

Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view 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 spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces 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 subspace (the analog of "dropping the altitude" of a triangle) 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 a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, the Hilbert space can also be usefully thought of in terms of the space of infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space. Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum.

## Definition and illustration

### Motivating example: Euclidean space

One of the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by 3, 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

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

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 ||x||, and to the angle θ between two vectors x and y by means of the formula

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

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

consisting of vectors in 3 is absolutely convergent provided that the sum of the lengths converges as an ordinary series of real numbers:[1]

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

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

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

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 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:

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

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:

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

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

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

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

A second example is the space 2 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

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

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

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

### 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.[2]

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:

• The inner product of a pair of elements is equal to the complex conjugate of the inner product of the swapped elements:
${\displaystyle \langle y,x\rangle ={\overline {\langle x,y\rangle }}\,.}$
• The inner product is linear in its first[nb 1] argument. For all complex numbers a and b,
${\displaystyle \langle ax_{1}+bx_{2},y\rangle =a\langle x_{1},y\rangle +b\langle x_{2},y\rangle \,.}$
${\displaystyle \langle x,x\rangle >0}$ where ${\displaystyle x\neq 0}$ and
${\displaystyle \langle x,x\rangle =0}$ where ${\displaystyle x=0\,.}$

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

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

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 bilinear: that is, linear in each argument.

The norm is the real-valued function

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

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

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

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:

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

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

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

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 pre-Hilbert space.[3] 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

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

converges absolutely in the sense that

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

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

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

converges. The inner product on l2 is defined by

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

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).[4]

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