# Has the existence of Hilbert Space been proven 100%?

Or is it only theoretical.

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

(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")

martinbn, bhobba and jim mcnamara
bhobba
Mentor
Its part of some models like negative numbers are part of some models. You cant have a negative number of ducks but if you owe someone some ducks it can be modelled as a negative number. Same with complex numbers. You cant have a complex electric current but for sinusoidal currents you can model it using complex numbers and much of the math simplifies if you do that.

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

Thanks
Bill

FactChecker
Gold Member
There are a great many examples of Hilbert spaces. They do exist. You might want to ask if a particular space you are interested in is a Hilbert space.

mathman
There are a great many examples of Hilbert spaces. They do exist. You might want to ask if a particular space you are interested in is a Hilbert space.
Hilbert space is a purely mathematical concept, generalized Euclidean space. Much of quantum theory uses Hilbert space as part of the development.

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

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

mathman
The real numbers and the complex plane are both Hilbert spaces.
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.

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.
The opening post asked if a Hilbert Space exist so I gave the simplest ones, and BTW:

Wikipedia
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.
Complete - ‎Tensor product of Hilbert spaces - ‎Categoryilbert space

Hilbert Space -- from Wolfram MathWorld
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.

[PDF]Hilbert Spaces - UC Davis Mathematics
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 ...

Hilbert spaces | Quantiki
https://quantiki.org/wiki/hilbert-spacesQuantiki
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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