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If x and y are represented in Cartesian coordinates, then the dot product is defined by

The dot product takes two vectors x and y, and produces a real number x ⋅ y. One of the most familiar examples of a Hilbert space is the Euclidean vector space consisting of three-dimensional vectors, denoted by R 3, and equipped with the dot product.

9 Orthogonal complements and projectionsĭefinition and illustration Motivating example: Euclidean vector space.

8.2 Bessel's inequality and Parseval's formula.

5.2 Parallelogram identity and polarization.

1.1 Motivating example: Euclidean vector space.The latter space is often in the older literature referred to as the Hilbert space. When this basis is countably infinite, it allows identifying the Hilbert space with the space of the infinite sequences that are square-summable. 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. 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. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. Geometric intuition plays an important role in many aspects of Hilbert space theory. 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. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. 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). 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. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. A Hilbert space is a vector space equipped with an inner product which defines a distance function for which it is a complete metric space. In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional.

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 state of a vibrating string can be modeled as a point in a Hilbert space.

Abstract colorful wave vector illustration