What is Hilbert space: Definition and 231 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. stevendaryl

    Question About Hilbert Space Convention

    According to Wikipedia:http://en.wikipedia.org/wiki/Hilbert_space the inner product \langle x | y \rangle is linear in the first argument and anti-linear in the second argument. That is: \langle \lambda_1 x_1 + \lambda_2 x_2 | y \rangle = \lambda_1 \langle x_1 | y \rangle + \lambda_2 \langle...
  2. Clear Mind

    Are these quantum states equivalent in Hilbert space representation?

    I've started few days ago to study quantum physics, and there's a thing which isn't clear to me. I know that a quantum state is represented by a ray in a Hilbert space (so that ##k \left| X \right\rangle## is the same state of ##\left| X \right\rangle##). Suppose now to have these three states...
  3. N

    Ising model, Hiblert space, Hamiltonian

    Can anyone please explain to me what is the Ising model, Hilbert space, and Hamiltonian ? However, please explain it as simple as possible because I am a freshman. I have looked up all three things. I've tried my best to make some sense of it, but I am, honestly, still confused on what any of...
  4. 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...
  5. M

    Finding the Normal Cone of a Closed Convex Subset in a Hilbert Space

    Let \text{ } H \text{ }be \text{ }a \text{ } Hilbert \text{ } space, \text{ }K \text{ }be \text{ }a \text{ }closed\text{ }convex\text{ }subset \text{ } of \text{ }H \text{ }and \text{ }x_{0}\in K. \text{ }Then \\N_{K}(x_{0}) =\{y\in K:\langle y,x-x_{0}\rangle \leq 0,\forall x\in K\} .\text{...
  6. K

    Requirement of Separability of Hilbert Space

    I have started reading formal definitions of Hilbert Spaces. I don't understand the requirement of separability postulate. I have proved that it leads to count ability of basis but again why is that required at first place.
  7. L

    Operators on infinite-dimensional Hilbert space

    Hello all! I have the following question with regards to quantum mechanics. If ##H## is a Hilbert space with a countably-infinite orthonormal basis ##\{ \left | n \right \rangle \}_{n \ \in \ \mathbb{N} }##, and two operators ##R## and ##L## on ##H## are defined by their action on the basis...
  8. P

    Canonical Commutation Relations in finite dimensional Hilbert Space?

    So lately I've been thinking about whether or not it'd be possible to have the commutation relation [x,p]=i \hbar in a Hilbert Space of finite dimension d. Initially, I was trying to construct a lattice universe and a translation operator that takes a particle from one lattice point to the...
  9. T

    Question about wavefunctions and their Hilbert space

    Maybe someone here can explain me something I never understood in QM: The wave function lives in the Hilbert space spanned by the measurement operator. Is there any mathematical relation of those spaces with each other?
  10. W

    Bra and Ket Representation in Dual Hilbert Space

    Hi, if ket is 2+3i , than its bra is 2-3i , my question is 2+3i is in Hilbert space than 2-3i can be represented in same hilbert space, but in books it is written we need dual Hilbert space for bra?
  11. S

    Determining how to find the 2-d hilbert space from fusing ising anyons

    Hello all, I'm working through the following paper on topological quantum computing. http://www.qip2010.ethz.ch/tutorialprogram/JiannisPachosLecture In particular I'm trying to derive and solve the pentagon equation in order to evaluate the F matrices for ising anyons. One thing I'm trying...
  12. A

    Hilbert space, orthonormal basis

    My book says that "the countability of the ONS in a hilbert space H entails that H can be represented as closure of the span of countably many elements". I must admit my english is probably not that good. At least the above quote does not make sense to me. What is it trying to say? Previously...
  13. A

    Orthonormal system in Hilbert space

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  14. K

    Spectrum of Momentum operator in the Hilbert Space L^2([-L,L])

    Homework Statement Find the spectrum of the Momentum operator in the Hilbert Space defined by L^2([-L,L]), consisting of all square integrable functions ψ(x) in the range -L, to L Homework Equations We can get the resolvent set containting all λ in ℂ such that you can always find a...
  15. L

    Do Hilbert Space Isomorphism Map Dense Sets to Dense Sets?

    Suppose that H, K are Hilbert spaces, and A : H -> K is a bounded linear operator and an isomorphism. If X is a dense set in H, then is A(X) a dense set in K? Any references to texts would also be helpful.
  16. G

    Given a Hamiltonian how do you pick the most convenient Hilbert space?

    For example, I have a 3D particle that experiences a harmonic oscillator potential only in the X,Y plane for all Z ie. a free particle in the Z direction. This seems like cylindrical coordinates but I'm not sure how to express the Hilbert space if I want to be able to describe states and...
  17. H

    Dimension of Hilbert space (quantum mechanics)

    Homework Statement Consider the states with the quantum numbers n = l = 1 and s = 1/2 Let J = L + S What is the dimension of the Hilbert space to describe all states with these quantum numbers? Homework Equations The Attempt at a Solution I believe the dimension of the Hilbert...
  18. A

    The pure-point subspace of a Hilbert space is closed

    (All that follows assumes we are talking about a self-adjoint operator A on a Hilbert space \mathscr H.) The first volume of Reed-Simon defines \mathscr H_{\rm pp} = \left\{ \psi \in \mathscr H: \mu_\psi \text{ is pure point} \right\}. The book seems to take for granted that \mathscr H_{\rm...
  19. L

    Do bras and inner products relate in a Rigged Hilbert Space?

    One of the most important results of functional analysis is that for every bounded linear functional f: H → ℂ on a Hilbert space H, there exists a fixed |v> in H such that f(|u>) is equal to the inner product of |v> with |u> for all |u> in H. This justifies the labeling of f as <v| in the...
  20. P

    A problem about non-separable Hilbert space

    also see http://planetmath.org/exampleofnonseparablehilbertspace the main difficulty is about the completeness, which is hard to prove, the author's hint seems don't work here, for you can not use the monotone convergence theorem directly , f(x)χ[-N,N]/sqrt[N] is not monotone
  21. I

    MHB Proving H is Complete & a Hilbert Space: Analysis of $\|.\|_H$

    Hi, Let H = \{(x_n)_n \subseteq \mathbb{R} | \sum_{n=1}^{\infty} x_n < \infty \} and for $(x_n)_n \in H$ define $$\|(x_n)_n\|_H = \sup_{n} \left|\sum_{k=0}^{n} x_k \right|$$ Prove that $H$ is complete. Is $H$ a Hilbert space? What is the best way to prove $H$ is complete? To prove it's a...
  22. K

    Hilbert Space in Quantum Mechanics

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  23. C

    Finite Hilbert Space v.s Infinite Hilbert Space in Perturbation Theory

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  24. M

    Dimension of Rays in Hilbert Space

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  25. B

    Verifying Inner Product & Showing $\ell^{2}$ is a Hilbert Space

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  26. T

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  27. P

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    Is something wrong in my assertions below? Suppose we have two quantum systems N and X. Let N is described by discrete observable \hat{n} (bounded s.a. operator with discrete infinite spectrum) with eigenvectors |n\rangle. Let X is described by continuous observable \hat{x} (unbounded s.a...
  28. P

    Rigged Hilbert space, separable space, domain of CSCO, mapping

    Suppose that we have rigged Gilbert space Ω\subsetH\subsetΩ\times (H is infinite-dimensional and separable). Is the Ω a separable space? Is the Ω\times a separable space? Consider the complete set of commuting observables (CSCO) which contain both bounded and unbounded operators...
  29. C

    Problem with changing basis in Hilbert space

    The expansion theorem in quantum mechanics states that a general state of a system can be represented by a unique linear combination of the eigenstates of any Hermitian operator. If that's the case then that would imply we would be able to represent the spin state of a particle in terms of...
  30. C

    How Does Trace Class Operator Q Relate to Norms in Hilbert Spaces?

    Hello, I'm reading Gaussian measures on Hilbert spaces by S. Maniglia (available via google) and I have the following issue, regarding the proof. He states in Lemma 1.1.4: Let μ be a finite Borel measure on H. Then the following assertions are equivalent: (1) \int_H |x|^2 \mu(dx) < \infty (2)...
  31. C

    Can unitary operators on hilbert space behaive like rotations?

    Homework Statement unitary operators on hilbert space Homework Equations is there a unitary operator on a (finite or infinite) Hilbert space so that cU(x)=y, for some constant (real or complex), where x and y are fixed non-zero elements in H ? The Attempt at a Solution I know the...
  32. B

    Equivalent vectors in a Hilbert space

    In Griffith's intro to QM it says on page 95 (in footnote 6) : "In Hilbert space two functions that have the same square integral are considered equivalent. Technically, vectors in Hilbert space represent equivalence classes of functions." But that means that if we take for example...
  33. E

    Square integrable functions - Hilbert space and light on Dirac Notation

    Square integrable functions -- Hilbert space and light on Dirac Notation I started off with Zettilis Quantum Mechanics ... after being half way through D.Griffiths ... Now Zettilis precisely defines what a Hibert space is and it includes the Cauchy sequence and convergence of the same ... is...
  34. S

    A compact, bounded, closed-range operator on a Hilbert space has finite rank

    Homework Statement Let H be an \infty-dimensional Hilbert space and T:H\to{H} be an operator. Show that if T is compact, bounded and has closed range, then T has finite rank. Do not use the open-mapping theorem. Let B(H) denote the space of all bounded operators mapping H\to{H}, K(H) denote...
  35. J

    Hilbert space formulation for non-quantum mechanics?

    Note: I am NOT talking about the classical limit of quantum mechanics, where in the limit of numbers that are large compared to h the average values approach the classical values, nor am I talking about Lagrangin/Hamiltonian mechanics in phase space; I am talking about using vectors with...
  36. Rasalhague

    Rigged Hilbert Space: Algebraic v.s. Continuous Dual Space

    Definitions of a rigged Hilbert space typically talk about the "dual space" of a certain dense subspace of a given Hilbert space H. Do they mean the algebraic or the continuous dual space (continuous wrt the norm topology on H)?
  37. M

    What is the differene between Rigid Hilbert Space and Hilbert Space?

    What is the difference between Rigid Hilbert Space and Hilbert Space?
  38. A

    The polarization identity in Hilbert space

    If we assume the inner product is linear in the second argument, the polarization identity reads (x,y) = \frac 14 \| x + y \|^2 - \frac 14 \| x - y \|^2 - \frac i4 \|x + iy\|^2 + \frac i4 \| x - iy \|^2. But there is another identity that I've seen referred to in some texts as the...
  39. L

    Understanding the Hilbert Space Postulate in Quantum Mechanics?

    So one of the postulate of quantum mechanics is that observables have complete eigenfunctions. Can someone let me know if I am understanding this properly: Basically you postulate for example, position kets |x> such that any state can be represented by a linear combination of these states...
  40. W

    Span of a linearly independent subset of a hilbert space is a subspace iff finite

    Homework Statement Let S be a linearly independent subset of a Hilbert space. Prove that span(S) is a subspace, that is a linear manifold and a closed set, if and only if S is finite. Homework Equations The Attempt at a Solution Assuming S is finite means that S is a closed set...
  41. A

    Exploring the Hilbert Space of QFT

    Hi everyone. Many texts when describing QFT start immediately discussing about free field theories, Fock spaces etc.. I want to understand general properties of the Hilbert space, and how to find a basis of it, and how to find a particle interpretation. I know there are very mathematical...
  42. L

    Hilbert Space and Tensor Products

    I never thought about this stuff much before, but I am getting confused by a couple of things. For example, would the state ∣E,l,m⟩ be the tensor product of ∣E⟩, ∣l⟩, ∣m⟩, ie. ∣E⟩∣l⟩∣m⟩? I always just looked at this as a way to keep track of operators that had simultaneous eigenfunctions in a...
  43. L

    Hilbert Space and Tensor Product Questions.

    I never thought about this stuff much before, but I am getting confused by a couple of things. For example, would the state ∣E,l,m⟩ be the tensor product of ∣E⟩, ∣l⟩, ∣m⟩, ie. ∣E⟩∣l⟩∣m⟩? I always just looked at this as a way to keep track of operators that had simultaneous eigenfunctions in a...
  44. sunrah

    What is the optimal polynomial of degree 2 to minimize the given expression?

    Homework Statement P_{2} \subset L_{2} is the set of all polynomials of degree n \leq 2. Complete the following approximation. In other words find the polynomial of degree 2 that minimises the following expression: \int \left|cos(\frac{\pi t}{2}) - p(t)\right|^{2}dt = min with -1 <= t >= 1...
  45. L

    Questions about Rigged Hilbert Space

    Dirac's bra-ket formalism implicitly assumed that there was a Hilbert space of ket vectors representing quantum states, that there were self-adjoint linear operators defined everywhere on that space representing observables, and that the eigenvectors of any such operator formed an orthogonal...
  46. K

    Weak convergence of orthonormal sequences in Hilbert space

    So, I've found the result that orthonormal sequences in Hilbert spaces always converge weakly to zero. I've only found wikipedia's "small proof" of this statement, though I have found the statement itself in many places, textbooks and such. I've come to understand that this property follows...
  47. LarryS

    Scope of Hilbert Space of a System?

    In QM a system is represented by a Hilbert Space rather than a classical Phase Space. So, system A might be described by Hilbert Space Ha and system B might be described by Hilbert Space Hb. Mathematically, Hilbert Spaces are many things, but the first thing they are, at the most fundamental...
  48. K

    When did Hilbert Space First Enter the Undergrad Math Curriculum?

    How long does it take for newly discovered math material or physics material to be standardized into the math undergrad curriculum? Just wondering about hilbert space as well. When did hilbert space first go into the undergrad curriculum?
  49. L

    Does every Hilbert space have an identity?

    I am sure that my questions are stupid. If we have a Hilbert space H, what do we mean by the closed subspace of H. Also, Does every Hilbert space have an identity? :P. Could anyone please clean to me these things . Thanks!
  50. nomadreid

    Why does the inner produce in Hilbert space use fg* and not f*g?

    The inner product is supposed to give the probability amplitude for state u turning into state v. Taking a little mini-universe of two dimensions in the complex plane if I rotate vector u by multiplying it by scalar a to get vector v, then I end up that a = u*v/(||u||.||v||); given that the...
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