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

    Hi there, In his book "Quantum field theory and the standard model", Schwartz assumes that the canonical commutation relations for a free scalar field also apply to interacting fields (page 79, section 7.1). As a justification he states: I do not understand this explanation. Can you please...
  7. H

    Bounded operators on Hilbert spaces

    I have to show that for two bounded operators on Hilbert spaces ##H,K##, i.e. ##T \in B(H)## and ##S \in B(K)## that the formula ##(T \bigoplus S) (\alpha, \gamma) = (T \alpha, S \gamma)##, defined by the linear map ##T \bigoplus S: H \bigoplus K \rightarrow H \bigoplus K ## is bounded...
  8. A

    A Hilbert spaces and kets "over" manifolds

    Background: One can construct a Hilbert space "over" ##\mathbb{R}^{3}## by considering the set of square integrable functions ##\int_{\mathbb{R}^{3}}\left|\psi(\mathbf{r})\right|^{2}<\infty##. That's what is done in QM, and there, even if they are not normalizable, to every...
  9. R

    I How do you normalize this wave function?

    I have a basic question in elementary quantum mechanics: Consider the Hamiltonian $$H = -\frac{\hbar^2}{2m}\partial^2_x - V_0 \delta(x),$$ where ##\delta(x)## is the Dirac function. The eigen wave functions can have an odd or even parity under inversion. Amongst the even-parity wave functions...
  10. AndreasC

    Quantum Hilbert spaces and quantum operators being infinite dimensional matrices

    I just realized quantum operators X and P aren't actually just generalizations of matrices in infinite dimensions that you can naively play with as if they're usual matrices. Then I learned that the space of quantum states is not actually a Hilbert space but a "rigged" Hilbert space. It all...
  11. nomadreid

    I Quantum logic based on closed Hilbert space subspaces

    One proposal that I have read (but cannot re-find the source, sorry) was to identify a truth value for a proposition (event) with the collection of closed subspaces in which the event had a probability of 1. But as I understand it, a Hilbert space is a framework which, unless trivial, keeps...
  12. B

    Dimension of Hilbert spaces for identical particles

    My thoughts are: a) it should just be N^2 b) just N since they're identical c) due to Pauli exclusion would it be N^2 - N since they have to be different states?
  13. V

    A Equivalence Relation to define the tensor product of Hilbert spaces

    I'm following this video on how to establish an equivalence relation to define the tensor product space of Hilbert spaces: ##\mathcal{H1} \otimes\mathcal{H2}={T}\big/{\sim}## The definition for the equivalence relation is given in the lecture vidoe as ##(\sum_{j=1}^{J}c_j\psi_j...
  14. A

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

    Hi all, Just a clarification question as I'm learning. It's possible to have Feynman diagrams where the internal lines (virtual particles) are in fact on shell. 'On shell' would imply 'observable,' (maybe?) but as noted in @A. Neumaier's great FAQ, only sets of Feynman diagrams have predictive...
  15. J

    A Reproducing Kernel Hilbert Spaces

    I have been reading a lot about Reproducing Kernel Hilbert Spaces mainly because of their application in machine learning. I do not have a formal background in topology, took linear algebra as an undergrad but mainly have encountered things such as, inner product, norm, vector space...
  16. M

    Uncovering the Relationship Between Norms and Hilbert Spaces

    Homework Statement Given a Hilbert space ##V## and vectors ##u,v\in V##, show $$\|u-4v\| = 2\|u-v\| \iff \| u \| = 2 \| v\|.$$ Homework Equations The parallelogram identity $$2\| x \|^2+2\| y \|^2 = \| x-y \|^2 + \| x+y \|^2$$ The Attempt at a Solution Forward: $$\|u-4v\| = 2\|u-v\|...
  17. fresh_42

    Insights Hilbert Spaces And Their Relatives - Part II - Comments

    Greg Bernhardt submitted a new PF Insights post Hilbert Spaces And Their Relatives - Part II Continue reading the Original PF Insights Post.
  18. fresh_42

    Insights Hilbert Spaces and Their Relatives - Comments

    Greg Bernhardt submitted a new PF Insights post Hilbert Spaces and Their Relatives Continue reading the Original PF Insights Post.
  19. SemM

    I Exploring the Equivalence of Different Representations in Quantum Mechanics

    Hi, I found this article very interesting, given the loads of question I have posted in this regard in the last months. I cannot recall where I got the link from, and if it came from Bill Hobba in some discussion, thanks Bill! If not, thanks anyway for your answers and contributions. Here is...
  20. bhobba

    A Rigged Hilbert Spaces In Quantum Mechanics

    In discussing stuff in another thread I used the standard Dirac notion expanding a state in position eigenvectors namely |u> = ∫f(x) |x>. By definition f(x) is the wave-function. I omitted the dx which is my bad but the following question was posed which I think deserved a complete answer. It...
  21. I

    I Interpretation of direct product of Hilbert spaces

    Dear all, I know how to interpret a vector, inner product etcetera in one Hilbert space. However, I can not get my head around how the direct product of two (or more) Hilbert spaces can be interpreted. For instance, the Hilbert space ##W## of a larger system is spanned by the direct product of...
  22. F

    I Why are Hilbert spaces used in quantum mechanics?

    In classical mechanics we use a 6n-dimensional phase space, itself a vector space, to describe the state of a given system at anyone point in time, with the evolution of the state of a system being described in terms of a trajectory through the corresponding phase space. However, in quantum...
  23. B

    Vectors, Hilbert Spaces, and Tensor Products

    If I ever say anything incorrect, please promptly correct me! The state of a system in classical mechanics is specified by point in phase space, the point giving us the position and velocity at a given instance. Could we rephrase it by saying a vector in phase space specifies the system? If...
  24. Andre' Quanta

    Separability of Hilbert Spaces

    Why we require the separability of Hilbert spaces in Quantum Mechanics?
  25. W

    Commutation between operators of different Hilbert spaces

    Hi! If I have understood things correctly, in a multi-electron atom you have that the spin operator ##S## commutes with the orbital angular momentum operator ##L##. However, as these operators act on wavefunctions living in different Hilbert spaces, how is it possible to even calculate the...
  26. pellman

    How does this statement follow? (adjoints on Hilbert spaces)

    If A is an operator on a Hilbert space H and A* is its adjoint, then . That is, the orthogonal complement of the range of A is the same subspace as the kernel of its adjoint. Then the author I am reading says it follows that the statements "The range of A is a dense subspace of H" and "A* is...
  27. B

    Are Hilbert spaces still necessary?

    I have never been happy with the fact a single quantum state could be encoded by an infinite number of vectors |\phi\rangle. Choosing a unit vector limits this overabundance but you have still an infinity of (physically equivalent) possibilities left. I later realized that the projector...
  28. A. Neumaier

    When are isomorphic Hilbert spaces physically different?

    In quantum mechanics, a Hilbert space always means (in mathematical terms) a Hilbert space together with a distinguished irreducible unitary representation of a given Lie algebra of preferred observables on a common dense domain. Two Hilbert spaces are considered (physically) different if this...
  29. W

    Special Properties of Hilbert Spaces?

    Hi All, AFAIK, the key property that separates/differentiates a Hilbert Space H from your generic normed inner-product vector space is that , in H, the norm is generated by an inner-product, i.e., for every vector ##v##, we have## ||v||_H= <v,v>_H^{1/2} ##, and a generalized version of the...
  30. carllacan

    Energy levels and Hilbert Spaces

    Hi. Is there a Hilbert Space for each energy level of a system? (And, in general, for every point in time?) I read in some book that if a equation for a problem accepts two different sets of wavefunction solutions (the case in question was the free particle and the sets of solutions in...
  31. nomadreid

    Dimensions of Hilbert Spaces confusion

    If I understand it, Hilbert spaces can be finite (e.g., for spin of a particle), countably infinite (e.g., for a particle moving in space), or uncountably infinite (i.e., non-separable, e.g., QED). I am wondering about variations on this latter. The easiest uncountable to imagine is the...
  32. S

    Quantum Mechanics Book and resources on Hilbert Spaces

    I am currently in a modern physics course and would to do more advanced study in quantum mechanics before taking the senior-level Quantum Mechanics course at my school. We use Townsend's modern physics book for the class that I am in right now; here is a link...
  33. Y

    Tensor product of Hilbert spaces

    Hi everyone, I don't quite understand how tensor products of Hilbert spaces are formed. What I get so far is that from two Hilbert spaces \mathscr{H}_1 and \mathscr{H}_2 a tensor product H_1 \otimes H_2 is formed by considering the Hilbert spaces as just vector spaces H_1 and H_2...
  34. B

    Non-separable Hilbert spaces

    What goes wrong if you try to do QM/QFT with a non-separable Hilbert space? Why do the Wightman axioms stipulate a separable space? And I need something else cleared up: The Hilbert space of non-trivial QFTs are indeed non-separable right?
  35. J

    What good are rigged hilbert spaces?

    I have seen the definition of a rigged Hilbert space many times, but I have never seen rigged Hilbert spaces actually being used for anything. Like for proving something. Has anyone else ever seen rigged Hilbert spaces being used for proving something?
  36. K

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

    Homework Statement Let H be a Hilbert space. Prove \Vert x \Vert = \sup_{0\neq y\in H}\frac{\vert (x,y) \vert}{\Vert y \Vert} The Attempt at a Solution First suppose x = 0. Then we have \sup_{0\neq y\in H}\frac{\vert (x,y) \vert}{\Vert y \Vert} = \sup_{0\neq y\in H}\frac{\vert (0,y)...
  37. T

    How are Banach and Hilbert spaces applied in quantum mechanics?

    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

    A question regarding tensor product of hilbert spaces

    So I've been reading Cohen-Tannoudji's "Quantum mechanics vol 1" and have understood the part that proves that the hilbert space of a 3-dimensional particle can be described/decomposed as a tensor product of hilbert spaces using position vectors (or analogously momentum vectors) in the x, y and...
  39. T

    On the use of Hilbert Spaces to represent states

    There are a lot of coherent definitions of Hilbert spaces. Let's take the wikipedia one and let me ask you some cuestions: A Hilbert space: 1) Should be linear 2) Should have an inner product (the bra - ket rule to get an amplitude) 3) Should be complete (every cauchy sequence should be...
  40. I

    Question about the Bolzano Weierstrass analogue in Hilbert spaces

    While reading a proof on the closure of the span of finite number vectors in a hilbert space with respect to the norm induced topology, I became stumped on a particular step of the proof using the Bolzano Weierstrass theorem. For finite dimensional vector spaces, Bolzano Weierstrass states...
  41. S

    Hilbert Spaces And Quantum Mechanics

    While going through the hermitian nature of quantum operators i came upon the term hilbert spaces... i have no idea what so ever what does this means and would like to know that what are Hilbert spaces and what are they doing in quantum mechanics...
  42. R

    Need book suggestion: Introduction to Hilbert Spaces

    Our last course on Mathematical Physics covers topology, topological spaces, metric spaces; differential forms; introduction to group theory including finite and continuous groups, group representations, and Lie groups. The textbook to be used is Math methods by Arfken and Intro to Hilbert...
  43. A

    Exploring the Differences between H1 & H2 Hilbert Spaces

    Hilbert Space... Can somebody tell me the difference between H1 & H2 hilbert spaces?
  44. MathematicalPhysicist

    Proving the Continuity of Norms in Hilbert Spaces for q>=p

    Homework Statement Prove that for q>=p and any f which is continuous in [a,b] then || f ||_p<=c* || f ||_q, for some positive constant c. Homework Equations The norm is defined as: ||f||_p=(\int_{a}^{b} f^p)^\frac{1}{p}. The Attempt at a Solution Well, I think that because f is...
  45. N

    What is the difference between Hilbert Spaces and other metric spaces?

    I've been reading about them (briefly), and can't see any large difference between them and metric spaces or even euclidean spaces for that matter. What am I missing? I read a Hilbert Space is a complete inner product space. But a metric space is a complete space as well with the only...
  46. diegzumillo

    Are Hilbert spaces uniquely defined for a given system?

    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

    Projections on Banach and Hilbert spaces

    I've now encountered two different definitions for a projection. Let X be a Banach space. An operator P on it is a projection if P^2=P. Let H be a Hilbert space. An operator P on it is a projection if P^2=P and if P is self-adjoint. But the Hilbert space is also a Banach space, and there's...
  48. C

    Hilbert spaces: Explicit example of an unbounded operator?

    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

    So, say I have a composite hilbert space H = H_A \otimes H_B, can I write any operator in H as U_A \otimes U_B? Thanks
  50. R

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

    {T_a} is an orthonormal system (not necessarily countable) in a Hilbert space H. x is an arbitrary vector in H. i must show that the inner product <x, T_a> is different fron 0 for at most countably many a. i'm not even quite sure where to begin. i know that the inner product is the...