Hilbert Space

A Hilbert Space is a complete normed vector (Banach) space that has a norm induced by an inner product.

AKA: Complete Inner Product Space, Infinite-Dimensional Euclidean Space.
Context:
- It can typically provide Hilbert Space Completeness through hilbert space cauchy sequence convergence.
- It can typically enable Hilbert Space Inner Product Structure through hilbert space angle and length measurement.
- It can typically support Hilbert Space Orthogonal Decomposition through hilbert space perpendicular projections.
- It can typically facilitate Hilbert Space Fourier Analysis through hilbert space orthonormal basis expansions.
- It can typically implement Hilbert Space Norm Relationships through hilbert space inner product induced norms.
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- It can often enable Hilbert Space Kernel Methods through hilbert space reproducing kernels.
- It can often support Hilbert Space Machine Learning through hilbert space kernel-based algorithms.
- It can often implement Hilbert Space Quantum Mechanics through hilbert space quantum state representation.
- It can often provide Hilbert Space Matrix Analysis through hilbert space linear operators.
- It can often facilitate Hilbert Space Distance Metrics through hilbert space induced metrics.
- It can often enable Hilbert Space Optimization through hilbert space variational methods.
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- It can range from being a Finite-Dimensional Hilbert Space to being an Infinite-Dimensional Hilbert Space, depending on its hilbert space dimensional structure.
- It can range from being a Separable Hilbert Space to being a Non-Separable Hilbert Space, depending on its hilbert space basis cardinality.
- It can range from being a Real Hilbert Space to being a Complex Hilbert Space, depending on its hilbert space scalar field.
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- It can contain Hilbert Space Projections for hilbert space geometric transformations.
- It can support Hilbert Space Operators for hilbert space linear transformations.
- It can utilize Hilbert Space Orthonormal Basises for hilbert space vector representations.
- It can satisfy Hilbert Space Parallelogram Law through hilbert space inner product induced norm.
- It can implement Hilbert Space Riesz Representation through hilbert space linear functional duality.
- It can be used to create mathematical framework systems
- It can support infinite dimensional geometric analysis tasks).
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Examples:
- Hilbert Space Types, such as:
  - Finite-Dimensional Hilbert Spaces, such as:
    - Euclidean Vector Space for hilbert space finite dimensional geometry.
    - Complex Coordinate Space for hilbert space finite complex analysis.
  - Infinite-Dimensional Hilbert Spaces, such as:
    - L2 Space for hilbert space square integrable functions.
    - l2 Space for hilbert space square summable sequences.
- Hilbert Space Structures, such as:
- Hilbert Space Applications, such as:
  - Reproducing Kernel Hilbert Space (RKHS)s, such as:
    - Gaussian RKHS for hilbert space machine learning kernel.
    - Polynomial RKHS for hilbert space algebraic function space.
  - Hilbert Space Kernel Methods, such as:
    - Kernel-based Learning Algorithm using hilbert space embedding.
    - Kernel Function providing hilbert space inner product.
  - Hilbert Space Quantum Systems, such as:
    - Quantum State as hilbert space vector.
    - Quantum Observable as hilbert space self-adjoint operator.
  - Hilbert Space Matrix Analysiss, such as:
    - Symmetric Real-Number Matrix for hilbert space self-adjoint operator.
    - Normal Matrix for hilbert space spectral decomposition.
  - Hilbert Space Distance Metrics, such as:
    - Euclidean Distance Metric in hilbert space finite dimensional case.
    - Frobenius Norm for hilbert space matrix operator norm.
  - Hilbert Space Mathematical Analysiss, such as:
    - Mathematical Analysis Task using hilbert space functional analysis.
    - Singular Value decomposition in hilbert space operator theory.
    - Growth Function analysis in hilbert space learning theory.
  - Hilbert Space Information Theorys, such as:
    - Normalized Mutual Information Metric in hilbert space statistical analysis.
    - Random Element in hilbert space probability theory.
  - Hilbert Space Theorems, such as:
    - Mercer's Theorem for hilbert space kernel representation.
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Counter-Examples:
- Banach Space, which provides complete normed vector space without hilbert space inner product structure.
- Inner Product Space, which offers inner product without hilbert space completeness requirement.
- Normed Vector Space, which lacks both hilbert space inner product and hilbert space completeness.
See: Inner Product Space, Topological Vector Space, Vector Space, Mathematical Space, Reproducing Kernel Hilbert Space, Kernel-based Learning Algorithm, Quantum State, Kernel Function, Mercer's Theorem.

References

2025a

(Wikipedia, 2025) ⇒ "Hilbert space". In: Wikipedia. Retrieved: 2025-05-31.
- QUOTE: "Hilbert space is a real or complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The inner product allows lengths and angles to be defined. Furthermore, completeness means that there are enough limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space."

"Hilbert spaces are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis, and ergodic theory. Examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces, and Hardy spaces of holomorphic functions."

"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."

2025b

(MathWorld, 2025) ⇒ "Hilbert Space". In: Wolfram MathWorld. Retrieved: 2025-05-31.
- QUOTE: A Hilbert space is a complete inner product space. The most familiar examples are the finite-dimensional Euclidean spaces, but the concept extends to infinite-dimensional spaces of functions. The norm in a Hilbert space is induced by the inner product, and the distance between two points is defined in terms of this norm.

2021

(MIT OCW, 2021) ⇒ MIT OpenCourseWare. (2021). "Lecture 14: Hilbert Spaces". In: 18.102 Introduction to Functional Analysis.
- QUOTE: A Hilbert space is a vector space equipped with an inner product that is complete with respect to the norm defined by that inner product. The concept of orthogonality, projection, and basis in Hilbert spaces generalizes familiar notions from Euclidean geometry to more abstract settings.

2015

(Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Hilbert_space Retrieved:2015-6-7.
- 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, this means that the Hilbert space can also usefully be thought of in terms of infinite sequences that are square-summable. 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.

2012

(Venkatachalaiyer, 2012) ⇒ Alampallam Balakrishnan Venkatachalaiyer. (2012). “Introduction to Optimization Theory in a Hilbert Space." Vol. 42. Springer Science & Business Media,
- QUOTE: … A normed linear space is complete if every Cauchy sequence converges (in norm) to an element in the space. Def. A Hilbert Space is a complete inner product space. We note that every normed linear space can be completed. …

2004

(Lanckriet et al., 2004a) ⇒ Gert R. G. Lanckriet, Nello Cristianini, Peter Bartlett, Laurent El Ghaoui, and Michael I. Jordan. (2004). “Learning the Kernel Matrix with Semidefinite Programming.” In: The Journal of Machine Learning Research, 5.
- QUOTE: Kernel-based learning algorithms (see, for example, Cristianini and Shawe-Taylor, 2000; Scholkopf and Smola, 2002; Shawe-Taylor and Cristianini, 2004) work by embedding the data into a Hilbert space, and searching for linear relations in such a space. The embedding is performed implicitly, by specifying the inner product between each pair of points rather than by giving their coordinates explicitly.

1997

(Luenberger, 1997) ⇒ David G. Luenberger. (1997). “Optimization by Vector Space Methods." Wiley Professional. ISBN:047118117X

1980

(Reed & Simon, 1980) ⇒ Michael Reed & Barry Simon. (1980). "Hilbert Spaces". In: Functional Analysis (Cambridge University Press).
- QUOTE: "A Hilbert space is a complete inner product space, central to functional analysis and quantum mechanics. The geometry of Hilbert spaces allows for the generalization of concepts such as orthogonality, projection, and Fourier series to infinite-dimensional settings."

1970

(Pollingher & Zaks, 1970) ⇒ Adolf Pollingher, and Abraham Zaks. (1970). “On Baer and Quasi-Baer Rings." Duke Mathematical Journal 37, no. 1
- QUOTE: … generated by an idempotent. The motivation comes from the observation that the theory of rings of operators on a Hilbert space is a particular case of a pure algebraic theory of Baer rings satisfying some axioms. …

1964

(Nussbaum, 1964) ⇒ A E Nussbaum. (1964). “Reduction Theory for Unbounded Closed Operators in Hilbert Space." Duke Mathematical Journal 31, no. 1
- QUOTE: … matrix o] the operator A. Since a projection P in a Hilbert space is a bounded self-adjoint idempotent operator, it follows that P is a projection in 3C X 3C if and only if the elements of the matrix (P.) satisfy the relations (i) P*=P. and …

1955

(Bram, 1955) ⇒ Joseph Bram. (1955). “Subnormal Operators." Duke mathematical journal 22, no. 1
- QUOTE: … Throughout this paper, a Hilbert space is a vector space over the complex numbers, an operator is a bounded linear transformation, and a subspace is a closed linear manifold. ...