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

learn more… | top users | synonyms

0
votes
0answers
16 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
votes
1answer
12 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
votes
0answers
30 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
votes
0answers
66 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
votes
1answer
29 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 $\|\cdot\| = (\cdot, \cdot)^{\frac{1}{2}}$). ...
0
votes
2answers
58 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
votes
1answer
34 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
votes
2answers
46 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
votes
0answers
24 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
votes
1answer
30 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
votes
2answers
32 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 ...
-2
votes
2answers
55 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
89 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
28 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
vote
1answer
34 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
16 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
votes
2answers
68 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
votes
0answers
52 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
votes
1answer
17 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
votes
1answer
127 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
votes
1answer
48 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
vote
1answer
28 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
vote
1answer
26 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.
0
votes
0answers
15 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
votes
1answer
71 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
votes
0answers
28 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 ...
3
votes
0answers
63 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
votes
1answer
30 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
votes
1answer
28 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
votes
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: ...
-1
votes
1answer
30 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
46 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
votes
0answers
18 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 ...
1
vote
1answer
36 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
votes
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
vote
0answers
58 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 $\sum_n|b_n|^2<\infty$
1
vote
1answer
25 views

Counterexample for Palais-Smale condition

I have trouble proving that functional $I:H\to\mathbb{R}$ given by $$I(u)=\frac{1}{2}\|u\|^2-\frac{1}{2}(u,f)^2$$ does not satisfy Palais-Smale condition if $\|f\|=1$. I managed to prove that when ...
1
vote
1answer
25 views

distributivity of tensor product and direct sum for Hilbert spaces

Before I ask my actual question about direct sums and tensor products of Hilbert spaces, let's first talk about direct sums and tensor products of vector spaces. We might define direct sums of ...
1
vote
1answer
27 views

Equivalent Definitions of the $L_2$ inner product.

If $g \in L_2(\mathbb{R})$, then we can define the $L_2$ norm to have the following relationship: $\|g\|_2^2 = \int_\mathbb{R} g^2$. If $A\subseteq \mathbb{R}$, then we can define the norm of $L_2(A)$ ...
2
votes
1answer
37 views

Canonical isometric isomorphism of $l_{\alpha}^{2}$

Let $\alpha \in \mathbb{R}$ and $l_{\alpha}^{2}$ the vector space of bi-infinite sequences $(x_{n})_{n\in \mathbb{Z}}$ such that $||x||_{\alpha}:=\sum_{n\in\mathbb{Z}} ...
1
vote
0answers
21 views

Orthonormal Basis of $L^{2}(0,1)$?

Are the functions $e_n := e^{i\cdot(2n+1)\cdot\pi\cdot x}$, $n \in \mathbb{Z}$ an Orthonormal basis of $L^{2}(0,1)$? I suppose it is true, but I haven't been able to prove it myself yet.
1
vote
1answer
72 views

Weak convergence in $C[0,1]$

For a uniformly bounded sequence $(f_n)$ in $C[0,1]$, show that $f_n$ converges weakly to $0$ $\iff $ $\lim \limits_{n \to \infty} f_n(y) =0$ for all $y \in [0,1]$ Is the equivalence true if we do ...
0
votes
1answer
21 views

Continuity of bounded and convex function on Hilbert space

I'm looking for the proof (or at least hints how to prove it) of theorem: Let $H$ be a real Hilbert space and let $f:H\to (-\infty,+\infty]$ be a convex function, bounded from above in some ...
1
vote
1answer
28 views

Equivalent Definitions of Negative Order Sobolev Spaces

Ignoring fractional sobolev spaces, if we restrict ourselves to $k>0$ when $k$ is an integer, then the Sobolev space of order $k$, for $W^{k,p}(\mathbb{R})$ is the space of functions $f$ such that ...
0
votes
0answers
19 views

Relationship between $T$ and $T^{*}$ for $T\in B(H)$

Let $T\in B(H)$ for a Hilbert space $H$. Then consider $T^{*}\in B(H)$, the dual operator. I'm trying to verify the relationship $\text{ker}(T)^{\perp} = \overline{\text{Im}(T^{*})}$ For the ...
4
votes
2answers
208 views
+100

Optimization problem using Reproducing Kernel Hilbert Spaces

I am encountering a problem concerning Reproducing Kernel Hilbert Spaces (RKHS) in the context of machine learning using Support Vector Machines (SVMs). With refernce to this paper [Olivier Chapelle, ...
2
votes
0answers
40 views

Fractional Brownian motion---construction via Hilbert space?

The Brownian motion has the following (Levy-Ciesielski?) construction via Hilbert space isomorphisms: Let $\{ Z_i \}_{i \in \mathbb{Z}}$ be i.i.d. $N(0,1)$ random variables defined on $(\Omega, ...
0
votes
1answer
41 views

Spectrum of the operator $A(f,g)=(g,\Delta f-f)$

Let $\Omega$ be an open set in $\mathbb{R^n}$. We consider the product Hilbert space $H=H^1_0(\Omega)\times L^2(\Omega)$ with the norm $$|(f,g)|^2=\int_\Omega (|\nabla f|^2+|f|^2+|g|^2 ) dx$$ We ...
1
vote
0answers
36 views

Any finite-dimensional subspace of a Hilbert space is closed: easier proof?

A noted theorem is that a finite-dimensional subspace of a Hilbert space must be topologically closed. I have seen some proofs of this theorem which are less simple than this, but what is wrong with ...
0
votes
0answers
22 views

Question about convergence of sum

Let $T\in B(H,E)$ where $H$ a seperable hilbertspace, $E$ a seperable Banach space. By parsevals identity $$\left\|T^*\right\|^2= \sup_{ \left\|x^*\right\|\leq 1}\left\|T^*x^*\right\|^2 = \sup_{ ...