The mathematical concept of a Hilbert space generalizes the notion of Euclidean space. It extends the methods of Euclidean geometry and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces of any finite or infinite dimension. A Hilbert space is an abstract vector space, and it has the additional structure of an inner product that allows length and angle to be measured. Finally, Hilbert spaces are required to be complete, a property that stipulates the existence of enough limits in the space to allow the techniques of calculus to be used. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert (after whom they are named), 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 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 R 3 {\displaystyle \mathbf {R} ^{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:
( x 1 x 2 x 3 ) ⋅ ( y 1 y 2 y 3 ) = x 1 y 1 + x 2 y 2 + x 3 y 3 . {\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:
It is symmetric in x and y: x ⋅ y = y ⋅ x. It is linear in its first argument: (ax1 + bx2) ⋅ y = a(x1 ⋅ y) + b(x2 ⋅ y) for any scalars a, b, and vectors x1, x2, and y. 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
x ⋅ y = ‖ x ‖ ‖ y ‖ cos θ . {\displaystyle \mathbf {x} \cdot \mathbf {y} =\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|\,\cos \theta \,.}
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
∑ n = 0 ∞ x n {\displaystyle \sum _{n=0}^{\infty }\mathbf {x} _{n}}
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