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Or is it only theoretical.

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In summary, Hilbert space is a mathematical concept that exists in the same way as sets, vectors, or the number line. It is not just a theoretical concept, as it has practical applications in fields such as quantum mechanics. Hilbert spaces are used to model complex systems and are a common tool in mathematical analysis. They are defined as complete inner product spaces and examples of finite-dimensional Hilbert spaces include the real numbers and the complex plane. In conclusion, Hilbert space is a well-established concept in mathematics with numerous examples and practical uses.

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Or is it only theoretical.

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Hilbert space is a mathematical concept. It exists the same way that sets exist, or vectors, or the number line.KarminValso1724 said:Or is it only theoretical.

(This also might be a good time for you to learn what the word "theoretical" means. It does not mean "not proven", "speculative", "something we aren't yet sure about")

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Hilbert spaces are like that - useful in some models particularly QM.

Thanks

Bill

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Hilbert space is a purely mathematical concept, generalized Euclidean space. Much of quantum theory uses Hilbert space as part of the development.FactChecker said:

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Yes. Hilbert spaces are more the rule than the exception in spaces that we study. There are examples everywhere. The only reasonable question is whether a particular unusual space is a Hilbert space. So the OP should specify what space he is asking about.mathman said:Hilbert space is a purely mathematical concept, generalized Euclidean space. Much of quantum theory uses Hilbert space as part of the development.

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The real numbers and the complex plane are both Hilbert spaces.KarminValso1724 said:Or is it only theoretical.

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Although it might simply be a matter of definition, but Hilbert space is usually defined as an infinite dimensional analog of n dimensional Euclidean space.Zafa Pi said:The real numbers and the complex plane are both Hilbert spaces.

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The opening post asked if a Hilbert Space exist so I gave the simplest ones, and BTW:mathman said:Although it might simply be a matter of definition, but Hilbert space is usually defined as an infinite dimensional analog of n dimensional Euclidean space.

Wikipedia

The mathematical concept of a Hilbert space, named after David Hilbert,

Complete - Tensor product of Hilbert spaces - Categoryilbert space

mathworld.wolfram.com › ... › Mathematical HumorMathWorld

by EW Weisstein - 2004 - Cited by 3 - Related articles

A Hilbert space is a vector space with an inner product such that the norm defined by. turns into a complete metric space. If the metric defined by the norm is not complete, then is instead known as an inner product space. Examples of finite-dimensional Hilbert spaces include.

https://www.math.ucdavis.edu/.../ch6.pdfUniversity [Broken] of California, Davis

Definition 6.2 A Hilbert space is a complete inner product space. In particular, every Hilbert space is a Banach space with respect to the norm in. (6.1). Example ...

https://quantiki.org/wiki/

In mathematics, a '''Hilbert space''' is an inner product space that is complete with respect to the norm defined by the inner product. Hilbert spaces serve to clarify ...

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