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. quasar987

    Little bit of convex analysis on a Hilbert space

    [SOLVED] Little bit of convex analysis on a Hilbert space Homework Statement Let H be a Hilbert space over R and f:H-->R a function that is bounded below, convex and lower semi continuous (i.e., f(x) \leq \liminf_{y\rightarrow x}f(y) for all x in H). (a) For all x in H and lambda>0, show that...
  2. quasar987

    Hilbert space & orthogonal projection

    [SOLVED] Hilbert space & orthogonal projection Homework Statement Let H be a real Hilbert space, C a closed convex non void subset of H, and a: H x H-->R be a continuous coercive bilinear form (i.e. (i) a is linear in both arguments (ii) There exists M \geq 0 such that |a(x,y)| \leq...
  3. quasar987

    Convergence in Hilbert space question

    Homework Statement Is it true/possible to show that in a Hilbert space, if z_n is a sequence (not known to converge a priori) such that (z_n,y)-->0 for all y, then z_n-->0 ? The Attempt at a Solution I've shown that if z_n converges, then it must be to 0. But does it converge?
  4. quasar987

    The Hilbert space L²([0,2pi], R) and Fourier series.

    Homework Statement Is there a way to prove that E={sin(nx), cos(nx): n in N u {0}} is a maximal orthonormal basis for the Hilbert space L²([0,2pi], R) of square integrable functions (actually the equivalence classes "modulo equal almost everywhere" of the square integrable functions)? I...
  5. P

    Hilbert Space: Closest point property

    Homework Statement The theorem about the closest point property says: If A is a convex, closed subspace of a hilbert space H, then \forall x \in H\,\, \exists y \in A:\,\,\,\, \| x-y\| = \inf_{a\in A}\|x-a\| I have to show that it is enough to show this theorem for x = 0 only, by...
  6. Z

    Exploring \psi(x) in Hilbert Space

    [SOLVED] Hilbert Space Homework Statement For What Values of \psi(x)=\frac{1}{x^{\alpha}} belong in a Hilbert Sapce?Homework Equations \int x^{a}=\frac{1}{a+1} x^{a+1} The Attempt at a Solution I tried to use the condition that function in Hilbert space should satisfy: \int\psi^{2}=A but it...
  7. 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...
  8. marcus

    Noncommutative Geometry blog has audio of David Hilbert

    the NCG blog has various interesting stuff one thing was this link to a 4 minute talk by David Hilbert http://math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.mp3 wide audience, nontechnical here's NCG blog http://noncommutativegeometry.blogspot.com/ and the brief post about...
  9. C

    Why Hilbert Space is Necessary for Quantum Mechanics

    It's fairly well known that "the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors") residing in a complex separable Hilbert space", and that a Hilbert space is basically just the generalisation "from the two-dimensional plane and...
  10. 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...
  11. S

    Complete basis for quantum oscillator Hilbert space

    In the creation/annihilation operator picture the Hilbert state of a quantum harmonic oscillator is spanned by the eigenstates |n> of the number operator. I've never seen a proof that: 1. the ground state |0> is unique 2. the states |n> form a complete basis i.e. any state in that Hilbert...
  12. N

    Proving Density of U and V in Hilbert Space: Tips and Tricks

    Homework Statement Let e_i = (0,0,\ldots, 1, 0 , \ldots) be the basis vectors of the Hilbert space \ell_2^\infty. Let U and V be the closed vector subspaces generated by \{ e_{2k-1}|k \geq 1 \} and \{ e_{2k-1} + (1/k)e_{2k} | k \geq 1 \}]. Show U \oplus V dense in \ell_2^\infty I am...
  13. R

    Showing S1+S2 is Dense in Hilbert Space

    Homework Statement i have {ej} is an orthonormal basis on a hilbert space S1 is the 1-dimensional space of e1 and S2 is the span of vectors ej + 2e(j+1) eventually i need to show that S1 + S2 is dense in H and also evaluate S2 for density and closedness Homework Equations i know...
  14. S

    Olutions to Hilbert Hotel's Infinite Room Challenge

    Homework Statement There are infinite rooms in Hilbert Hotel, room number is natural number 0, 1, 2, Story: AhQ comes into Hilbert Hotel, but the waiter Kong Yiji tells him that all rooms are booked up. AhQ is disappointed. If you were the waiter, what would you do? 2. If there...
  15. T

    Understanding Einstein-Hilbert Action: Detailed Explanation from an Expert

    Can someone explain to me why this is the appropriate action?It makes some sens that that would be used, but I'd like a detailed explanation from someone familiar with the topic. Why is it the one that yields the proper equations?
  16. N

    Eigenfunctions in Hilbert Space, Infinite Square Wells and Uncertainty

    Hi I'm kinda stuck with a couple quantum HW questions and I was wondering if you guys could help. First, Is the ground state of the infinite square well an eigenfunction of momentum?? If so, why. If not, why not?? Second, Prove the uncertainty principle, relating the uncertainty in...
  17. 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
  18. 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...
  19. I

    Einstein Hilbert Action

    Can someone explain to me what Einstein Hilbert action is? and how it relates to the variational principle? I appreciate any help that I can get!
  20. M

    Boundary Conditions and Hilbert Space

    Hey all, Last year, I took my university's undergraduate QM sequence. We mainly used Griffiths' book, but we also used a little of Shankar's. Anyway, I decided to go through Shankar's book this year, in a more formal treatment of QM. After the first chapter, I already have some questions that...
  21. I

    Proving V is a Hilbert Space: Facts 1 & 2

    Fact 1: we know that a closed subspace of a Hilbert Space is also a Hilbert Space. Fact 2: we know that the Sobolev Space H^{1} is a Hlbert space. How do I show that the space V:=\{v \in H^{1}, v(1) = 0\} is a Hilbert space? Is V automatically a closed subspace of H^{1}? How do I show this...
  22. W

    Hilbert space of direct products

    How come if all states in the representation space (of say rotations) have the same energy, Hilbert space can be written as a direct product space of these representation spaces?
  23. dextercioby

    A question on Hilbert space theory

    Let's say we have a self-adjoint, densly defined closed linear operator acting on a separable Hilbert space H A:D_{A}\rightarrow H Let \lambda be an eigenvalue of A and let \Delta_{A}\left(\lambda\right) = \{\left(A-\lambda \hat{1}_{H}\right)f, \ f\in D_{A}\} How do i prove...
  24. CarlB

    Is There a Quantum Mechanics Framework Without Hilbert Space?

    Is anyone out there working on a theory of elementary particles that is basic quantum mechanics without the Hilbert space? The reason I'm asking is because I found this article by B. J. Hiley: Algebraic Quantum Mechanics, Algebraic Spinors and Hilbert Space. The orthogonal Clifford algebra...
  25. E

    Diagonalizability (in Hilbert Spaces)

    Under what circumstances is a (linear) operator \mathcal{H} \to \mathcal{H} between a Hilbert space and itself diagonalizable? Under what circumstances does (number of distinct eigenvalues = dimension of H), i.e., there exists a basis of eigenvectors with distinct eigenvalues? Although I am...
  26. CarlB

    Orientation and Hilbert space bra / ket factoring.

    I've been thinking about the probability interpretation of quantum states. In the density matrix formalism, or in measurement algebra like Schwinger's measurement algebra, one makes the assumption that pure states can be factored into bras and kets, and that bras and kets can be multiplied...
  27. benorin

    Are All Hilbert Spaces Isomorphic to a Subspace of Another?

    The questions reads: If H1 and H2 are two Hilbert spaces, prove that one of them is isomorphic to a subspace of the other. (Note that every closed subspace of a Hilbert space is a Hilbert Space.) What I'm thinking is that every separable Hilbert space is isomorphic to L2. If I recall, a...
  28. benorin

    Does the Fourier Series of a Continuous Function Converge Uniformly?

    So I'm working this HW problem, namely Suppose f is a continuous function on \mathbb{R}, with period 1. Prove that \lim_{N\rightarrow\infty} \frac{1}{N}\sum_{n=1}^{N} f(\alpha n) = \int_{0}^{1} f(t) dt for every real irrational number \alpha. The above is for context. The hint says...
  29. A

    Proving the Equivalence of Cardinalities with Hilbert's Hotel

    Just come across this question on a problem sheet and it's got me rather confused! You have to prove that |[0,1]|=|[0,1)|=|(0,1)| without using Schroeder-Bernstein and using the Hilbert Hotel approach. After looking at the Hilbert Hotel idea I can't really understand how this helps! This...
  30. Cincinnatus

    Exploring the Hilbert Cube: Understanding its "Cubelike" Nature

    So, what exactly is "cubelike" about the hilbert cube? I think I am having trouble "visualizing" it. Is it just called that because it it homeomorphic to I^inf. ?
  31. J

    Understanding Hilbert Space: Representing & Calculating Length of Functions

    I'm trying to understand Hilbert spaces and I need a little help. I know that it's a vector space of vectors with an infinite number of components, but a finite length. My biggest question is: how is a Hilbert space used to represent a function? Is each component of the vector a point on the...
  32. benorin

    Understanding Hilbert Space to Bilinear and Quadratic Forms

    OK, so I've been there before, Hilbert Space that is. You know, infinite dimensional function space. At least I thought I had, that is until I started reading A Hilbert Space Problem Book by Halmos. So operator theory, right. What's are bilinear, sesquilinear, conjugate linear, ect. -...
  33. Oxymoron

    Does T^*T = I imply that T is an isometry on a Hilbert space?

    Let \mathcal{H} be a Hilbert space over \mathbb{C} and let T \in \mathcal{B(H)}. I want to prove that \|Tx\| = \|x\| \, \Leftrightarrow \, T^{\ast}T = I for all x \in \mathbb{H} and where I is the identity operator in the Hilbert space. Since this is an if and only if statement I began...
  34. G

    Hilbert Space Help: Showing Norm Does Not Satisfy Parallelogram Law

    How can I show that the space of all continuously differentiable functions on [a,b] denoted W[a,b] with inner product (f,g)=Integral from a to b of (f(x)*conjugate of g(x)+f'(x)*conjugate of g'(x)). Should I show that the norm does not satisfy the parallelogram law?
  35. R

    Uncountable infinite dimensional Hilbert space

    Does anybody know an example for a uncountable infinite dimensional Hilbert space?(with reference or prove).i know about Banach space:\L_{\infty} has uncountable dimension(Functional Analysis,Carl.L.Devito,Academic Press,Exercise(3.2),chapter I.).but it is not a Hilbert space. thank you.
  36. Oxymoron

    Exploring Hilbert Spaces: Questions & Answers

    A Hilbert Space is a complete inner product space. My first question: From the definition above, is it safe to say that every sequence in a Hilbert Space converges? And so can we say that Hilbert Spaces only contain Cauchy sequences? Second question: These 'sequences' that we talk about...
  37. E

    The completeness of Hilbert Space

    Can anyone guide me through or point me to a link of a proof that Hilbert space is complete? I am doing a paper on Hilbert space so I introduced some of its properties and now want to show it is complete.
  38. Oxymoron

    Smallest Norm in a Hilbert Space

    I have this problem which I want to do before I go back to uni. The context was not covered in class before the break, but I want to get my head around the problem before we resume classes. So any help on this is greatly appreciated. Question Suppose C is a nonempty closed convex set in a...
  39. Oxymoron

    Linear Operators on Hilbert Spaces

    Let U, V, W be inner product spaces. Suppose that T:U\rightarrow V and S:V\rightarrow W are bounded linear operators. Prove that the composition S \circ T:U\rightarrow W is bounded with \|S\circ T\| \leq \|S\|\|T\|
  40. S

    Question about weak convergence in Hilbert space

    The Question is as follows: let A be a bounded domain in R^n and Xm a series of real functions in L^2 (A). if Xm converge weakly to X in L^2(A) and (Xm)^2 converge weakly to Y in L^2(A) then Y=X^2. i don't know if the above theorem is true and could sure use any help i can get. if...
  41. B

    Hilbert Space: Understanding and Exploring the Mathematical Concept

    hilbert space?? hai, what is hilbert space ?any important links known to you regarding that?please send some links .
  42. K

    How Do We Choose the Right Hilbert Space in Quantum Mechanics?

    Hi everyone, This summer (it's summer in Australia) I have been studying quantum mechanics from a mathematician's perspective, and the physical interpretation has become a little more difficult as the theory has become more in-depth. Do we have a particular method for choosing which...
  43. S

    Understanding Hilbert Space in Quantum Mechanics: A Beginner's Guide

    In quantum mechanic the Hilbert space is often used. I don't study physics, must be said. So, I have a few questions to this space (I can calculate with complex numbers and vectors). 1. What's the different between a bra <p| and a ket |b> vector? 2. What calculation is behind that: <p|b>...
  44. S

    Hilbert space and infinite norm vectors

    Quickly can we define a hilbert space (H, <,>) where the vectors of this space have infinite norm? (i.e. the union of finite + infinite norm vectors form a complete space). If yes, can you give a link to a paper available on the web? If no, can you briefly describe why? Thanks in advance...
  45. humanino

    Is a Hilbert space over the quaternions separable?

    Please forgive this physicist's thread : I can define a Hilbert space that is : 1) \mathbb{R}^n with the euclidian norm, especially on a real field, and which is finite dimensional : is it right ? This is the most stupid question ever. 2) over the quaternions \mathbb{H} ? 3) if the...
  46. E

    Hilbert Space,Dirac Notation,and some other stuff

    Ok, so I am a little unsure of how to apply these new concepts I am learning. Here is a question. The function g(x)=x(x-a)e^ikx is in a certain Hilbert space where the finite norm squared equals the integral of the product of Psi's complex conjugate and Psi (dx) is less than infinity...
  47. M

    Hilbert Spaces & Quantum Mechanics: Algebra, Geometry & Analysis

    I browsed a book by Byron & Fuller "Math. Physics" and read the following: Algebra, Geometry & Analysis are joined when functions are treated as vectors in a vector space. This makes Hilbert spaces extremely useful in QM.(paraphrased but that's the jist of it) Comments on this? If it's...
  48. enigma

    What is a Hilbert Transform and How Does it Work?

    Anybody know what a Hilbert transform does? The NB4 function is looking at how the frequency of noise from a gearbox changes as a damaged tooth passes the sensor. I understand the concept, but I don't understand what the math is actually computing...
  49. S

    Are there practical applications for l2 in mathematics and physics?

    Hi all. I heard about the site the the Sec Web. Anyway, I'm a math major with philosophy and physics minors, and I'm going to grad school in math this fall. I'm doing my senior seminar on l2, the set of square summable sequences. I'd like to close my paper and my talk on some applications...
  50. marcus

    Separable Hilbert Space in Loop Quantum Gravity

    Yesterday Meteor found this new paper of Rovelli's and added it to the "surrogate sticky" collection of links. In case there is need for discussion, it should probably have its own thread as well. The paper expands a key assertion made on page 173 of the on-line draft of Rovelli's book...
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