What is Hilbert spaces: Definition and 57 Discussions

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 a vector space equipped with an inner product, an operation that allows lengths and angles to be defined. Furthermore, Hilbert spaces are complete, which means that 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, the Hilbert space can also be usefully thought of in terms of the space of infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space. 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.

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  1. cianfa72

    I Quantum particle's state in momentum eigenfunctions basis

    Hi, as discussed in this recent thread, for a particle without spin the quantum state of the particle is described by a "point" in the Hilbert space of the (equivalence classes) of ##L^2## square-integrable functions ##|{\psi} \rangle## defined on ##\mathbb R^3##. The square-integrable...
  2. cianfa72

    I ##L^2## square integrable function Hilbert space

    Hi, I'm aware of the ##L^2## space of square integrable functions is an Hilbert space. I believe the condition to be ##L^2## square-integrable actually refers to the notion of Lebesgue integral, i.e. a measurable space ##(X,\Sigma)## is tacitly understood. Using properties of Lebesgue integral...
  3. M

    Homework Help: Quantum Information Theory 1

    I considered an operator ##X \in \mathcal{L}(\mathcal{X} \otimes \mathcal{K})##, that is positive, ##X \geq 0##. And I defined it as it follows: ##X = \sum_{i,j} a_{ij} ∣x_i \rangle \langle x_i ∣ \otimes ∣k_j \rangle \langle k_j∣ ## Where ##x_i## are basis for ##\mathcal{X}## and ##k_j## basis...
  4. Euge

    POTW Orthonormal Bases on Hilbert Spaces

    Let ##H## be a Hilbert space with an orthonormal basis ##\{x_n\}_{n\in \mathbb{N}}##. Suppose ##\{y_n\}_{n\in \mathbb{N}}## is an orthonormal set in ##H## such that $$\sum_{n = 1}^\infty \|x_n - y_n\|^2 < \infty$$ Show that ##\{y_n\}_{n\in \mathbb{N}}## must also be an orthonormal basis.
  5. J

    A POVMs for Infinite Dimensional Hilbert Spaces

    After reading up on some of the discussion in the Quantum Interpretations forums, I became interested in learning more about POVMs. However, most of the examples are from the finite dimensional setting. If I wanted to model a POVM that approximately measures position and momentum...
  6. E

    I Commutation relations for an interacting scalar field

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  7. H

    Bounded operators on Hilbert spaces

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  8. A

    A Hilbert spaces and kets "over" manifolds

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  9. R

    I How do you normalize this wave function?

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  10. AndreasC

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  11. nomadreid

    I Quantum logic based on closed Hilbert space subspaces

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  12. B

    Dimension of Hilbert spaces for identical particles

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  13. V

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  14. A

    I On-shell virtual particles and 'physical' Hilbert spaces....

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  15. J

    A Reproducing Kernel Hilbert Spaces

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  16. M

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  17. fresh_42

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  18. fresh_42

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  19. SemM

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  20. bhobba

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  21. I

    I Interpretation of direct product of Hilbert spaces

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  22. F

    I Why are Hilbert spaces used in quantum mechanics?

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  23. B

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  24. Andre' Quanta

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  25. W

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  26. pellman

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  27. B

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  28. A. Neumaier

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  29. W

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  30. carllacan

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  31. nomadreid

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  32. S

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  33. Y

    Tensor product of Hilbert spaces

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  34. B

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  35. J

    What good are rigged hilbert spaces?

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  36. K

    Proving the Norm of a Hilbert Space: Tips and Tricks for Success

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  37. T

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    As a european bachelor student in physics, i can follow a theoretical math course next year about banach and hilbert spaces. How useful are those subjects for physics?
  38. A

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  39. T

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  40. I

    Question about the Bolzano Weierstrass analogue in Hilbert spaces

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  41. S

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  42. R

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  43. A

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  44. MathematicalPhysicist

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  45. N

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  46. diegzumillo

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    Hi there! Repeating the question on the title: Are Hilbert spaces uniquely defined for a given system? I started to think about this when I was reading about Schrödinger and Heisenberg pictures/formulations. From my understanding, you can describe a system analyzing the time dependent...
  47. J

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  48. C

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    I'm wondering if someone can give me an example of an everywhere defined unbounded operator on a (separable for simplicity) Hilbert space in a "constructive" manner. Since it's unbounded, simply a dense definition (i.e. on an orthonormal basis) wouldn't work since you can't extend it by...
  49. A

    Composite Hilbert Spaces and Operators

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  50. R

    Proving <x, T_a> is Nonzero for Countable a in Hilbert Space H

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