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

  1. Apr 15, 2016 #1
    Or is it only theoretical.
     
  2. jcsd
  3. Apr 15, 2016 #2

    Nugatory

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    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")
     
  4. Apr 15, 2016 #3

    bhobba

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    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
     
  5. Apr 15, 2016 #4

    FactChecker

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    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.
     
  6. Apr 16, 2016 #5

    mathman

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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.
     
  7. Apr 16, 2016 #6

    FactChecker

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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.
     
  8. Apr 16, 2016 #7

    Zafa Pi

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    The real numbers and the complex plane are both Hilbert spaces.
     
  9. Apr 17, 2016 #8

    mathman

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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.
     
  10. Apr 18, 2016 #9

    Zafa Pi

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    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 - ‎Category:Hilbert 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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