In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.
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 vector 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 linear subspace or 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 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.
Definition and illustration
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 , and equipped with the dot product. The dot product takes two vectors and , and produces a real number . If and are represented in Cartesian coordinates, then the dot product is defined by
The dot product satisfies the properties
It is symmetric in and : . It is linear in its first argument: for any scalars , , and vectors , , and . 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. 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
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
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
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 , which is defined as the square root of the product of with its complex conjugate:
If is a decomposition of into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length:
The inner product of a pair of complex numbers and is the product of with the complex conjugate of :
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 the inner product of with another such vector is given by