Hilbert space Definition and 236 Threads

Homework Statement Let H be a Hilbert space. Prove that if \left\{ x _{n} \right\} is a sequence such that lim_{n\rightarrow\infty}\left\langle x_{n},y\right\rangle exists for all y\in H, then there exists x\in H such that lim_{n\rightarrow\infty} \left\langle x_{n},y\right\rangle =...

Can you define a space that is locally homeomorphic to an infinite dimensional Hilbert space analogously to how you define a (smooth) manifold by an atlas defining local homeomorphisms to R^n? So the charts would just map to the Hilbert space rather than R^n. Could the rest of the definition...

A question arose to me while reading the first chapter of Sakurai's Modern Quantum Mechanics. Given a Hilbert space, is the outer product \mathcal{H}\times \mathcal{H}^\ast \to End(\mathcal{H}); (| \alpha\rangle,\langle \beta|)\mapsto | \alpha\rangle\langle \beta| a surjection? Ie, can any...

[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...

[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...

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?

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...

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...

[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...

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...

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...

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...

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...

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...

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...

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...

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?

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...

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...

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...

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...

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...

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...

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. -...

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?

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.

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.

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...

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...

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

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...

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>...

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...

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...

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...

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...