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

1. Apr 15, 2016

### KarminValso1724

Or is it only theoretical.

2. Apr 15, 2016

### Staff: Mentor

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")

3. Apr 15, 2016

### Staff: 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

4. Apr 15, 2016

### FactChecker

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.

5. Apr 16, 2016

### mathman

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

6. Apr 16, 2016

### FactChecker

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.

7. Apr 16, 2016

### Zafa Pi

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

8. Apr 17, 2016

### mathman

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.

9. Apr 18, 2016

### Zafa Pi

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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