What is Hilbert: Definition and 302 Discussions

David Hilbert (; German: [ˈdaːvɪt ˈhɪlbɐt]; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory).
Hilbert adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set the course for much of the mathematical research of the 20th century.Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic.

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  1. 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)...
  2. 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)...
  3. 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...
  4. 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...
  5. 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?
  6. S

    Sequence in Hilbert space,example

    Hi. I`m working on some exercises but I could`t find any clue for this one: Find a bounded sequence (as like the norm) in l^2 Hilbert space,that weakly converges to 0 (as like the weak topology) but doesn`t have any convergent subsequences (as in strong topology). Could someone help me?
  7. 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...
  8. 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...
  9. N

    Hilbert action according to robert wald

    hi everyone i'm brand new to forums and I'm holdin a seminar on a variation of the hilbert action as described in wald's book general relativity. if anyone knows that book and topic pretty well maybe you can help me, my question is this: for the variation \delta R_{ab} with respect to...
  10. N

    [Number theory] Calculate the Hilbert symbol

    Homework Statement Determine the Hilbert symbol \left( \frac{2,0}{\mathbb F_{25}} \right) where the F denotes the field with 5² elements. Homework Equations \left( \frac{2,0}{\mathbb F_{5}} \right) = -1 The Attempt at a Solution Due to the formula that I put under "relevant equations"...
  11. 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...
  12. 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...
  13. 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...
  14. 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)?
  15. M

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

    What is the difference between Rigid Hilbert Space and Hilbert Space?
  16. H

    Proving \neg x \vee x in Hilbert System: A Logical Dilemma

    Homework Statement Prove \neg x \vee x using Hilbert system. Homework Equations The logical axioms. I'm not sure if I should state them, or whether there is a standard set. It seems to me that different sets are used. Anyway, the ones with disjunction in them are: a \rightarrow a \vee b...
  17. N

    Harmonic Functions, conjugates and the Hilbert Transform

    Hi, I am currently confused about something I've run across in the literature. Given that \nabla^2\phi = \phi_{xx}+\phi_{zz} = 0 for z\in (-\infty, 0] and \phi_z = \frac{\partial}{\partial x} |A|^2 at z=0. for A= a(x)e^{i \theta(x)} . The author claims that...
  18. 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...
  19. B

    Hilbert Spaces: Inner Product on R^3

    I have just realized that I accidently put it in wrong sub forum. This should be in 'calculus and beyond'. Homework Statement Prove the function <x,y>=x_1y_1+x_2y_2+x_3y_3 defines an inner product space on the real vector space R^3 where x=(x1,x2,x3) and y=(y1,y2,y3)The Attempt at a Solution...
  20. A

    Difference between hilbert space,vector space and manifold?

    Difference between hilbert space,vector space and manifold?? Physically what do they mean? I m really confused imagining them..Explanation with example would help me to understand there application ..THanks in advance
  21. 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...
  22. 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...
  23. 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...
  24. 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...
  25. 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...
  26. 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...
  27. 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...
  28. 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...
  29. 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...
  30. 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...
  31. 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?
  32. bhobba

    Rigged Hilbert Space's And Quantum Mechanics

    Ages ago when I first learned about QM the textbooks all said QM was formulated in a Hilbert Space. Yet you had this Dirac Delta function and infinite norms. Got a hold of Von Neumans book and yea it resolved it but by using the Stieltjes Integral and resolutions of the identity however that...
  33. 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!
  34. 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...
  35. A

    Writing PDEs as differential equations on Hilbert space

    Hi, I was reading a paper on control of the 1-D heat equation with boundary control, the equation being \frac{\partial u(x,t)}{\partial x}= \frac{\partial^2 u(x,t)}{\partial x^2} with boundary conditions: u(0,t)=0 and u_x(1,t)=w(t), where w(t) is the control input. The authors...
  36. P

    What is the Meaning of Hilbert Space?

    Please can anyone help me to display the meaning of Hilbert Space?
  37. N

    Going from system & observable to hilbert space & operator?

    Hello, Given a system and an observable that one wants to measure in it, how does on get the (or a?) relevant hilbert space and the suitable operator in it? The examples I've come across so far seem to rely on... well, I'd call it "vague reasoning", but the word 'reasoning' seems too much. It...
  38. A

    Densely defined linear operators on Hilbert space and their ranges

    Suppose T is an injective linear operator densely defined on a Hilbert space \mathcal H. Does it follow that \mathcal R(T) is dense in \mathcal H? It seems right, but I can't make the proof work... There is a theorem that speaks to this issue in Kreyszig, and also in the notes provided by my...
  39. Rasalhague

    Rigged Hilbert Space: Definition & Examples by de la Madrid (2005)

    Rafael de la Madrid writes: - de la Madrid (2005): "The role of the rigged Hilbert space in Quantum Mechanics" Could the second paragraph be restated as: "The elements of \Phi, the vectors, regarded as equivalence classes of functions differing only on sets of zero Lebesgue measure, can...
  40. snoopies622

    Is There a Metric Tensor in Hilbert Space That Transforms Vectors to Duals?

    When I was studying general relativity, I learned that to change a vector into a covector (or vice versa), one used the metric tensor. When I started quantum mechanics, I learned that the difference between a vector in Hilbert space and its dual is that each element of one is the complex...
  41. jfy4

    Physical Hilbert Space - Inner Product & Rigorous Account

    Hi, Could someone tell me, or refer me to a reference, about what the physical separable hilbert spaces are for the electroweak and strong forces. I'm looking for a defined inner product for the theories and a rigorous account of their hilbert spaces. Thanks a lot,
  42. A

    Finding an orthonormal basis for a reproducing kernel Hilbert space.

    Hello all, I'm currently working on a problem in which I'm attempting to characterize a centered Gaussian random process \xi(x) on a manifold M given a known covariance function C(x,x') for that process. My current approach is to find a series expansion $\xi(x) = \sum_{n=1}^{\infty} X_n...
  43. R

    Hilbert transform filter phase

    We know that hilbert transform is a linear filter whose frequency response is given as -j*sgn(f), where f is the baseband frequency. Hence magnitude response of this filter is 1 and phase response is -pi/2 for f > 0 and pi/2 for f < 0. Hence phase response curve is like a staircase function (...
  44. K

    Did David Hilbert beat Einstein to Relativity?

    Heard that from one of my math teachers. He said that they were both independently working on relativity but because einstein's mail got to the publishers first, he got the credit for it.
  45. L

    Elementary Hilbert space concepts/definitions.

    I'd appreciate it if anyone could help me clear up some concepts, the last chapter of one of my math courses is a (highly mysterious) introduction to Hilbert spaces (very very basic): What does it mean for a function to be "square-summable"? Has something to do with the scalar product in...
  46. V

    Quantum Mechanics without Hilbert Space

    Von Neumann developed the concept of Hilbert Space in Quantum Mechanics. Supposed he didn't introduce it and we didn't use Hilbert Space now. What are its counterpart in pure Schroedinger Equation in one to one mapping comparison? In details. I know that "the states of a quantum mechanical...
  47. 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...
  48. B

    Complex Hilbert Space as a Symplectic Space?

    Hi All: in the page: http://mathworld.wolfram.com/SymplecticForm.html, Complex Hilbert space, with "the inner-product" I<x,y> , where <.,.> is the inner-product Does this refer to taking the imaginary part of the standard inner-product ? If so, is I<x,y> symplectic in...
  49. W

    How Is the Unit Operator Derived in Quantum Many-Body Hilbert Spaces?

    I am reading J.W.Negele and H.Orland's book "Quantum Many-Particle Systems". I don't know how one can derive equation (1.40) on page 6. The question is For quantum many-body physics, suppose there are N particles. The hilbert space is H_{N}=H\otimesH\otimes...H. Its basis can be...
  50. 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...
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