Tagged Questions

For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.

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0answers
26 views

Convergence of this priori error in FEM?

Problem My attempt I think h is the size of the mesh. C is a constant which probably depends on the size of the mesh, I think. I think the error converges linearly and dependent on the size of ...
0
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1answer
28 views

Linear operator defined by its eigenvectors/values

Let $H$ be a Hilbert space, $(e_n)$ a complete orthonormal sequence, and $\lambda_n$ a bounded sequence of complex numbers. Let $A$ be defined such that the $(e_n)$ are the eigenvectors of $A$ and the ...
1
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1answer
47 views

Galerkin Orthogonality in this FEM?

Problem Galerkin orthogonality is but I am not sure if it is in the right form. How can you use this orthogonality here? I think I should expand the last inequality first somehow.
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0answers
23 views

Symmetricity of square-integrable functions respect to inner-products.

Let $f(x),g(x) \in \mathrm{L}_2$. If we define inner-product $(f(x),g(x)) = \int_a^b f^*g dx$, then prove that $(f,pg) = (pf,g)$, given $f(1) = Cf(0), C=e^{i\psi}, p=-i d/dx, a=0, b=1$. I tried ...
1
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1answer
26 views

Weighted $L_2$ Hilbert space

this is a question where I am trying to find a reference for a result but I haven't been able to find one at all. Define $L_2(\mathbb R,d\mu) = \{g\in \mathbb R: \int g^2d\mu <\infty\}$. I am ...
0
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1answer
44 views

Parseval's theorem to $\sum_{k=1}^\infty |\langle x,x_k\rangle|^2$.

Let $\{x_k\}$ be a collection of vectors in a Hilbert space. We take any $x\in H$. The symbol $\langle .,.\rangle$ denote the inner product. The question is as follows. I have to apply the Parseval's ...
4
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1answer
70 views

“Refinement” of the existence of faithful representations of C*-algebras?

Conway, in a course in operator theory, brings the statement 1. below as a theorem and statement 2. below as an exercise. Still, he states that 2. refines 1., but I can't see it. Every C*-algebra ...
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1answer
44 views

Are translates of Gaussians an overcomplete set in $L^2(\Bbb R)$?

Consider the Gaussian $\exp(-t^2/2)$. Is it the case that any function in $L^2(\Bbb R)$ can be written as a limit of a sum of scalings and translations of Gaussians? That is, for any $f\in L^2(\Bbb ...
2
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0answers
54 views

Trace class operators problem

Let $\mathcal{B}_1(\mathcal{H})$ be the set of trace class operators in a Hilbert space $\mathcal{H}$ and $\mathcal{H}^{(d)} = \bigoplus_{i=1}^d \mathcal{H}$ with $1 \leq d \leq \infty$. If $C \in ...
1
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2answers
66 views

The difference between a normed space being reflexive and being isomorphic to its dual

Quoting wikipedia "a normed vector space is reflexive if it coincides with its bidual". Another definition, more precise is that a normed vector space is reflexive if its evaluation map ...
4
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1answer
83 views

Composition of two orthogonal projections

Let $V$ be a finite dimensional Euclidean space and let $W_1,W_2$ be two subspaces of $V$. Let $P_1,P_2$ denote the projections onto $W_1,W_2$ respectively. Is it true that the composition $P_1\circ ...
0
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1answer
56 views

What is the smallest non-trivial Hilbert space?

I came to know without proof or explanation that smallest non-trivial Hilbart space is generated by two basis vectors. What is its proof? One example I know. Denote $a = (0 , 1)$ and $b = (1 , 0)$. ...
1
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1answer
73 views

Do there exist two vectors in a Hilbert space such that $(x,y)\geqslant k\|x-y\|^{-2}$?

Let $H$ be a Hilbert space, $(x,y)$ denote the inner product of the elements $x,y\in H$, $\|x\|$ denote the norm of $x\in H$, and $k>0$. Do there exist such $x,y\in H$ that $$ ...
2
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1answer
90 views

Uniform convex space

Please I want to know if this space $$H^1_{0,p}([0,+\infty))=\lbrace u, u\in AC([0,+\infty)), u(0)=u(+\infty)=0,\sqrt{p}u'\in L^2\rbrace$$ where $p>0$, $p\in L^1((0,+\infty))$ ...
0
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2answers
54 views

Complete ONS and pure point spectrum

In all that follows all operators are taken to be densely defined on a Hilbert space $H$. Some textbooks state that an operator $A$ on $H$ has pure point spectrum if $H$ admits a complete ONS (Hilbert ...
0
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1answer
13 views

For bounded operator $U$, show that if $UU^*$, $U^*U$ are projections, then $U$ is a partial isometry

Let $(\mathcal{H}, (\cdot, \cdot))$ be a Hilbert space over $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$. Suppose that $U : \mathcal{H} \to\mathcal{H}$ is a bounded linear operator such that $UU^*$ and ...
0
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0answers
39 views

Seeing that a function is a trigonometric polynomial

I'm working through Chapter 4 of Rudin's Real and Complex Analysis book right now, and I've found myself rather more confused than usual. In the proof of the completeness of the trigonometric system, ...
0
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0answers
131 views

Span of Dirac's delta distributions dense in Hilbert space of $L^2$ functions?

According to Wiki a set of elements of a Hilbert space(B) is a basis for that space if: Orthogonality: Every two different elements of $B$ are orthogonal: $⟨e_k,e_j⟩=0$ for all $k$, $j$ in $B$ with ...
0
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1answer
37 views

Question about different defintions of isometry on a Hilbert space

Let $(\mathcal{H} , (\cdot, \cdot))$ be a Hilbert space over the field $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$ (so the norm on $\mathcal{H}$ is given by $\|\cdot\| = (\cdot, \cdot)^{\frac{1}{2}}$). ...
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2answers
85 views

Looking for a book: $B(H)$ not reflexive

I'm looking for a book with a proof that for an infinite dimensional Hilbert space, $B(H)$ is not reflexive. Thank you.
2
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1answer
80 views

No trace on $B(H)$ if $H$ is infinite dimensional

Let $H$ be an infinite dimensional Hilbert space and $B(H)$ the bounded linear operators on $H$. Then thre is no ultra weakly continous non-zero positve trace $tr:B(H)\rightarrow \mathbb{C}$. I ...
0
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2answers
54 views

ONB: Density Check?

How to show that $\{\sin{kx}:k\in\mathbb{N}\}$ for $\{f\in\mathcal{L}^2[0,\pi]:f(0)=f(\pi)=0\}$ is an ONB? (Clearly they are orthogonal to each other but is their span also dense?) What general ...
0
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0answers
32 views

There is a unit sequence weakly converging to every element of unit ball

Suppose the Hilbert space $H$ has a countable (I assume Hilbert?) basis. Let $x \in H$ be such that $\lvert x \rvert \leq 1$. Show that there exists a sequence $\{u_{n}\}$ in $H$ with ...
2
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1answer
46 views

Closed subspace $M=(M^{\perp})^{\perp}$ in PRE hilbert spaces. [duplicate]

Is it true that $M=(M^{\perp})^{\perp}$ if $M$ is a closed subspace of a PRE hilbert space (a space with a scalar product, but that is not complete)? The proof of the analog fact for hilbert spaces ...
2
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2answers
42 views

Closed subspaces of $L^2(0,1)$

I would like to prove that the almost-everywhere constant functions, and the functions whose integral is 0 are closed subspaces of $L^2(0,1)$. It's readily seen that they are subspaces. I'm finding ...
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2answers
89 views

Group of Unitaries: Strong Continuity

Let $\mathcal{L}^2(\mathbb{R})$ be the the Hilbert space of square integrable functions, shortly $\mathcal{L}^2$. Consider the group of unitaries: $$U:\mathbb{R}\to ...
6
votes
1answer
126 views

Non-examples for the Kato-Rellich Theorem

The Kato-Rellich Theorem is a classical result stating that if $A,B$ are unbounded operators on a Hilbert space with $A$ self-adjoint, $B$ symmetric, $\mathcal D (A)\subset \mathcal D(B)$ and $$ ...
2
votes
2answers
45 views

Sequence of bounded linear operators implicating Cauchy sequence in $\mathbb K$

Let H be a Hilbert space and $(T_n)_{n \in \mathbb N}$ be a sequence in ${\rm BL}(H)$ (bounded linear operators) such that $(\langle y,T_nx \rangle)_{n \in \mathbb N}$ is a Cauchy sequence in $\mathbb ...
1
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1answer
44 views

Projecting self-adjoint operator onto closed subspace

Let $H$ be a complex Hilbert space and let $(Q, D(Q))$ be a closed, densely defined, positive semidefinite, Hermitian quadratic form on $H$. (That is, $D(Q)$ is a dense subspace of $H$, $Q$ maps ...
0
votes
1answer
24 views

Unique trace on a type $II_1$ von Neumann Algebra

Let $M \subseteq B(H)$ be a type $II_1$ von Neumann Algebra. Then any two non-zero ultraweakly continious normalised traces $Tr,tr : \rightarrow \mathbb{C}$ are equal. I'm trying to understand this ...
0
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2answers
69 views

Minimum of $f(x,y)=\sum_{n=0}^{+\infty}\frac{(n^2−nx−y)^2}{2^n}$

Show that $$f(x,y)=\sum_{n=0}^{+\infty}\frac{(n^2−nx−y)^2}{2^n}$$ is defined on $\Bbb{R}^2$, it has a minimum and find for which couple $(x, y)$ the minimum is reached. The first point is okay, ...
2
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0answers
56 views

Are these functions on a Hilbert space Lipschitz equivalent?

Let $H$ be a Hilbert space with inner product $\langle\cdot,\cdot\rangle$. Fix a bounded operator $T$ on $H$, and $1\leq p<\infty$ (you can assume $p$ is an integer if necessary). Consider the ...
0
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1answer
28 views

Minimal projections and Type II von Neumann Algebras.

Let $M \subseteq B(H)$ be a type $II_1$ factor. Can it contain a minimal projection? If it can't, what would go wrong? I assume something about the trace being faithful?
8
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1answer
140 views

How to Prove the Semi-parametric Representer Theorem

This question concerns the generalized Representer Theorem, due to Schölkopf, Herbrich, and Smola. In this magnificent work, the authors provide two versions of the Representer Theorem, a ...
2
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1answer
53 views

Implying a positive definite operator

If we are given that $A:V \rightarrow V$ is an operator where $V$ is a real Hilbert space. If we are given that $A$ is bounded, strictly positive $\big(\langle Au,u \rangle > 0$ for all $u \neq ...
1
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1answer
40 views

Operator Tensor Product

Let $S$ and $T$ be bounded operators over a Hilbert space $\mathcal{H}$. Define their tensor product $S\otimes T$ as acting on $\mathcal{H}\otimes\mathcal{H}$ by $S\otimes T(x\otimes y):=Sx\otimes Ty$ ...
1
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1answer
39 views

Polar decomposition in a von Neumann algebra

Let $M \subseteq B(H)$ be a von Neumann algebra and $T \in M$. If $T=U|T|$ is the polar decomposition of T, why is $U \in M$? I'm thinking it's because $M$ is SOT-closed, but I'm not entirely sure.
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0answers
16 views

H2 norm or H2 control probloem

I read a paper, it says: How to understand this definition of H2? I know the definition of the H2-norm of a system: or However, how to understand the paper one's from these formula or ...
0
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1answer
85 views

Momentum Operator: Selfadjoint Extensions

This might be a possible duplicate - please let me know if there is already a proof in another thread. Consider the momentum operator on $\mathcal{L}^2[0,2\pi]$: ...
0
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0answers
36 views

Tensor Product of Hilbert Spaces: incomplete?

Let $\mathcal{H}$ be an infinite dimensional Hilbert space and $\mathcal{H}\otimes_0\mathcal{H}$ its algebraic twofold tensor product. Define a scalar product on it as ...
4
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0answers
95 views

Positive Operators: Definition?

Let $A$ be a self adjoint element of a C*-algebra $\mathcal{A}$ resp. a self adjoint operator of the operator algebra $\mathcal{B}(\mathcal{H})$ of bounded operators over a Hilbert space ...
2
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1answer
31 views

Approximations of compact operators

Let $(\xi_n)_{n=1}^\infty$ be a sequence in a Hilbert space $K$ convergent to some $\xi$. Suppose we have a compact operator $T$ on $K$ such that $T\xi = 0$. Can we find a sequence of compact ...
0
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1answer
31 views

Positive-definite function on a group function on a group

I have quite a hard time understanding the definition of positive-definite functions that is based on Hilbert spaces, the one that I read from Wiki; it does not exactly specify that how $H$ relates to ...
2
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1answer
51 views

Composition of Partial Isometries

Let $H$ be a complex Hilbert space and $S,T \in B(H)$ partial isometries. Then $S T$ is a partial isometrie, if and only if $T^*(\ker(S)) \subseteq \ker(ST)$. Edit: My attempts so far: ...
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1answer
41 views

Number Operator closable on Fock Space?

In Bratelli Robinson the number operator in Fock space is defined as: $$\mathcal{D}(N):=\{\phi\in\mathcal{F}:\sum_{n=1}^\infty n|\|\phi_n\|<\infty\}\\ N:\mathcal{D}(N)\to ...
2
votes
2answers
58 views

Weak convergence of partial sums

I recently came across an interesting problem on weak convergence in $\ell^2 (\Bbb N)$. Suppose that we have canonical basis $\{e_i\}$ in $\ell^2 (\Bbb N)$. We need to prove that the sequence ...
2
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0answers
21 views

does the hilbert space construction of random variables allow for infinite variance?

I am reading a book (Hilbert Space Methods in Probability and Statistical Inference by Small) which says that random variables can be viewed as functions in the hilbert space $L^2$ with the inner ...
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1answer
41 views

Does there always exist an equivalent geometric problem for a given differential equation?

I recently came up with a method to show for any given differential equation whose solution is the power series there exists an equivalent geometric problem in Hilbert space. I was wondering if ...
0
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1answer
19 views

Question about a scalar product

It is well known that in a hilbert space $H$ with orthonormal basis $(e_n)_{n=1}^{\infty }$, we have for every $f, g \in H$ $$\displaystyle{\langle f,g\rangle=\sum_{n\in\mathbb{N}}\overline{ \langle ...
1
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0answers
59 views

how to prove $\sum_n |b_n|^2<\infty$

$\{b_n\}$ is a complex sequence, If for all $\ell^2$ sequences $a_n$, we have $\sum_n \bar{a}_nb_n$ converges . Prove that $\sum_n|b_n|^2<\infty$