Tagged Questions

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

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5
votes
3answers
77 views

If a linear operator has an adjoint operator, it is bounded

This is a question I'm struggling with for a while: Let $H$ be a Hilber space. Let $T,S: H\rightarrow H$ be linear operators (not neccessarily bounded) such that for every $x,y\in H$: $\langle ...
1
vote
1answer
40 views

Sequence in a hilbert space.

$M$ is a closed subspace of the Hilbert space $H$, and x $\in H$. Call $d = \inf_{y \in M} ||x - y||^2$ Show that there exist a sequence of elements $y_n$ of M such that $||y_n - x ||^2 \rightarrow ...
0
votes
1answer
15 views

Prove the operator on hilbert space is compact

My question is actually the same as the first part of this one, Prove that T is compact which has not been answered. I am thinking about two ways, 1) use a bounded sequence $\{g_n\}$, and try to ...
1
vote
1answer
26 views

If $T^{2}$ is a compact operator then $T$ is compact

Suppose $T$ is a bounded , self-adjoint operator on a Hilbert space such that $T^{2}$ is compact. Then prove that $T$ is compact. I proved it by continuous functional calculus but am looking for a ...
0
votes
0answers
6 views

Relation between RKHS and space of continuous functions

Consider a Mercer Kernel $K\colon \mathcal{X}\times \mathcal{X}\to \mathbb{R}$, $\mathcal{X}$ being a compact subset of $\mathbb{R}^m$, and its (unique) associated Reproducing Kernel HIlbert Space ...
3
votes
2answers
55 views

An orthonormal subset of a Hilbert space is closed.

In Rudin Real and Complex Analysis there is an exercise (6, Ch. 4) that asks to show that a countably infinite orthonormal set $\{u_n:n\in\mathbb{N}\}$ in a Hilbert space $H$ is closed and bounded but ...
1
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1answer
65 views

$\{x^nf(x)\}_{n\in\mathbb{N}}\subset L_2(a,b)$ as a complete system

I read in Kolmogorov-Fomin's (p. 430 here) the statement, sadly left without a proof, that if function $f:(a,b)\to\mathbb{C}$, measurable almost everywhere on $(a,b)$, where $-\infty\leq ...
3
votes
1answer
10 views

Question on Completeness of Derived Inner Product Space

Let $(\mathcal{H},\langle{,}\rangle)$ be a separable, infinite-dimensional Hilbert space. Let $\mathcal{X}''$ denote the space of bounded sequences in $\mathcal{H}$. For a Banach limit $L$, define a ...
2
votes
2answers
28 views

Find an approximation of the unit ball as a weak-limit of a sequence in the unit sphere

Let $H$ be an infinite dimensional Hilbert space. It is well known that the weak-closure of the unit ball is the unit sphere. But I want to prove it as basicaly as possible, using the ...
1
vote
1answer
34 views

How can I prove that a sequence such that every converging subsequence coverges to the same limit, converges?

I want to claim that if $(x_n)_{n\in N}$ is a sequence, and there is $a$ such that if $(x_{n_k})$ converges, so $\lim x_{n_k} = a$ (it means that all converging subsequences have the same limit), then ...
2
votes
1answer
19 views

Some closed subspace of $l_2$?

$(a)$ I was trying to define a continuous linear map $T$ on $l_2$ whose kernel would be the $A$ and can conclude $A=T^{-1}(0)$ and hence closed set? could anyone help me to solve any of one?
1
vote
0answers
32 views

Suppose $V$ is subspace of a Hilbert Space $\mathcal H$. Show the identity $\bar V = (V^{\bot})^{\bot}$

Suppose $V$ is subspace of a Hilbert Space $\mathcal H$. Show the identity $\bar V = (V^{\bot})^{\bot}$. I've already proved that if $U$ is a closed subspace then $U = (U^{\bot})^{\bot}$. I also ...
1
vote
1answer
58 views

Proving this set is dense in $\ell^2$

I found this weirdest question and was wondering how could this be proved. This question is a part of a beautiful semi-constructive built of two dense disjoint convex sets in $\ell^2$, which I find ...
0
votes
2answers
24 views

Unique nearest point property

Consider $\mathbb{R}^2$ with a norm defined by $\|(x,y)\| = |x|+|y|$. Define $\mathrm{dist}(K,p) = \inf_{q \in K} \|q-p\|$. Why are there infinitely many points $q \in K$ that satisfy $\|p-q\| = ...
0
votes
0answers
36 views

Uniquness and Exisstence of One Theorem

I need a short and nice Proof for Uniqueness and Existence of the following theorem: Suppose (H, <0,0> ) is a Hilbert space, and M is a closed convex set and $x \in H$, then there is a unique ...
2
votes
1answer
19 views

is there a convex bounded subset A of H such that A is not norm closed and A∩L is norm closed for every finite dimensional subspace L of H

"Given an infinite dimensional Hilbert space H. Show that there is a convex bounded subset A of H such that A is not norm closed and A∩L is norm closed for every finite dimensional subspace L of H." ...
0
votes
0answers
11 views

Show that there is a convex bounded subset A of H such that A is not norm closed and A∩L is norm closed for every finite dimensional subspace L of H. [on hold]

Given an infinite dimensional Hilbert space H. Show that there is a convex bounded subset A of H such that A is not norm closed and A ∩ L is norm closed for every finite dimensional subspace L of H.
1
vote
1answer
40 views
0
votes
0answers
6 views

proof of separability hilbert space H.

I think that a Hilbert space is called separable if it contains a complete orthonormal sequence. Finite dimensional Hilbert spaces are considered separable. I need to prove that the Hilbert space ...
0
votes
1answer
19 views

How to see injection and boundedness

Lemma. If $A$ is a bounded linear operator defined on a Hilbert space and $\|Af\| \geq c\|f\|$ and $\|A^*f\| \geq c\|f\|$ for some constant $c$. Then $A$ has a bounded inverse. In the proof of ...
2
votes
0answers
31 views

Equivalent formulation for compact operators

According to Wikipedia, an operator is compact if it can be written in the form $T(u)=\sum_{n=1}^\infty \lambda_n<f_n, u> g_n$, where $\{f_n\}$ and $\{g_n\}$ are orthonormal sets and ...
1
vote
0answers
23 views

$L^{2}(\mathbb{R})$ is a separable Hilbert space.

I want to show $L^{2}(\mathbb{R})$ is separable. My idea is $C_{c}(\mathbb{R})$ is dense in $L^{2}(\mathbb{R})$ in $L^2$ norm and polynomials with rational coefficients are dense in $C[a,b]$ in $\sup$ ...
0
votes
0answers
12 views

Hilbert space subspace, orthogonal projection

I do not know how to solve this problem, and I do no know why it has to be a closed subspace. Thank you very much.
1
vote
1answer
19 views

linear transformation between Hilbert space

By definition, $|T|=\sup|(Tf,g)|, |f|\le1,|g|\le1$ $$||T||\ge(Tf,f)$$ But I can not find an example such that $||T||>(Tf,f)$ for any $|f|<1$. Any suggestion? Thanks in advance~
4
votes
2answers
93 views

Is this following bilinear form coercive?

First of all I want to mention that this is homework, so don't spoil it and reveal all the answer. just some guidenss :) Let $H$ be a Hilbert space, $T:H\rightarrow H$ a bounded linear operator for ...
3
votes
1answer
59 views

Inequivalent norms (given by different inner products) on infinite dimensional Hilbert space.

I have this question in reviewing for my exam. Let $H$ be an infinite dimensional Hilbert space. Write down an inner product on $H$ that gives a norm inequivalent with the original norm. Is $H$ ...
0
votes
1answer
29 views

Two versions of Lax-Milgram theorem

I'm having some troubles differentiating between two versions of Lax-Milgram theorem, one shown in my class and one that I saw is common on the internet. Let $H$ be hilbert space, $B$ bilinear form ...
0
votes
2answers
22 views

Existence Adjoint Operator: Boundedness?

Context This would make the check on the GNS construction much more simple. Problem Given a Hilbert space $\mathcal{H}$. Consider a merely linear operator $A:\mathcal{H}\to\mathcal{H}$. Suppose ...
1
vote
1answer
44 views

perturbation by orthogonal projection

Let $G$ be an operator with discrete spectrum on Hilbert space $H$ such that $\ker G$ is different from $\{0\}$. Let $P$ be the orthogonal projection onto $\ker G$, and let $G_{0} = G+P$. My ...
1
vote
1answer
16 views

Finite sum $\sum_{r,k} p_kP_r(x_k)f(x_k)P_r(x_m)=f(x_m)$

Let $x_0,\ldots,x_n\in\mathbb{R}$ be $n+1$ arbitrary real points and $p_0,...,p_n>0$ be positive real numbers. Let $P_0,P_1,\ldots,P_n$ be polinomials such that $$\sum_{k=0}^n ...
1
vote
1answer
29 views

Partial Isometries: Characterizations

Any partial isometry satisfies: $$\Omega\Omega^*\Omega=\Omega$$ From this, one derives projections: $$\Omega^*\Omega,\Omega\Omega^*$$ Conversely, given projections: $$\Omega^*\Omega,\Omega\Omega^*$$ ...
0
votes
0answers
28 views

Scalar product of $L_2$ with $\mu(E):=\int_E gdx$

I read in Kolmogorov and Fomin's Элементы теории функций и функционального анализа (p. 400 here) that, if we define measure $\mu$ for $E\subset[-1,1]$ by $$\mu(E):=\int_E g(x)dx$$ where the integral ...
0
votes
1answer
19 views

Hilber transform on [0,1)

Let $\mathbb{T}=[0,1)$ and $H$ be a Hilbert transform on $L^p(\mathbb{T})$ when $2\leq p< \infty$. If $f$ is $L^p$ and $f_n$ is trignometric polynomial such that $f_n\rightarrow f$ in $L^p$ sense. ...
2
votes
1answer
36 views

Eigenvalues of an integral operator on $L^2[-1, 1]$

Find the eigenvalues of the integral operator $K: L^2[-1, 1] \to L^2[-1, 1]$ defined by $(Kx)(t) = \int_{-1}^1 (1 - 3t \tau)x(\tau) d\tau$. I began with the fact that eigenvalues must be values ...
1
vote
1answer
39 views

Orthogonal Complements of polynomials in $L^2[0,1]$

I have two very difficult questions in my home work in function analysis, in which I have two calculate the complements of the following sets, in $L^2[0,1]$: All polynomials in the variable $x^2$ ...
0
votes
0answers
18 views

$f$ square-summable on $X'\times X''$, $\varphi_m$ square-summable on $X'$ and $\int f\cdot\bar{\varphi}d\mu'$ square-summable on $X''$

Let $X:=X'\times X''$ be the product of measure spaces $(X',\mu')$ and $(X',\mu'')$, endowed with the Lebesge extension $\mu:=\mu'\otimes\mu''$ of product measure $\mu'\times \mu''$ defined by ...
2
votes
1answer
49 views

Bound the norm of the partial trace of an operator on a Hilbert space

Let $H=H_1 \otimes H_2$ a composite Hilbert space and let $A, B$ bounded linear operators on $H$, and we can assume they are trace class. Let $A_2$ we denote the operator on $H_2$ obtained by taking ...
1
vote
1answer
19 views

Convolution Operator and Integration Operator

I have some questions about the following two operators. A convolution operator $T$. If $k \in \mathcal L^1(\mathbb R)$, then $$f(x) \mapsto \int_{-\infty}^\infty k(x-y)f(y) dy: \mathcal L^2(\mathbb ...
2
votes
1answer
43 views

Closed Subspaces of Hilbert Spaces

I read the following statements. But I do not know how to show it or any example to support it. Could anyone provide some explanation and examples, please? Thank you! The subspace $C^\infty$ ...
1
vote
0answers
21 views

Basis for Finite Dimensional Hilbert Spaces

Verify that a Hilbert space orthonormal basis in a finite dimensional Hilbert space is the same as an orthonormal basis in the sense of linear algebra. Here is what I know. Hilbert space ...
0
votes
1answer
37 views

Bounded Linear functional is the orthogonal projection onto its range.

Suppose we have $P:H\to H$, where $H$ is a hilbert space and $P$ is bounded and linear. Assume that it satisfies $P^2=P$ and $P^*=P$ where $P^*$ is the adjoint. Show that $||P||\leq 1$, that ...
0
votes
1answer
44 views

Møller Operators: Unitary Equivalence

Reference This is taken out of M. Reed and B. Simon, Scattering Theory. Problem Given a Hilbert space $\mathcal{H}$. Consider a free Hamiltonian $H_0$ and a perturbed one $H$. Introduce the ...
0
votes
0answers
12 views

totality of subset of an inner product space

How can I show that, given a subset $M$ of an inner product space $X$: If $M$ is a total set, then, $M^\perp =\{0\}$? ($M^\perp $ is the orthogonal complement of $M$)
-1
votes
1answer
27 views

Møller Operators: Absolutely Continuous Subspaces [closed]

Given a Hilbert space $\mathcal{H}$. Consider a free Hamiltonian $H_0$ and a perturbed one $H$. Introduce the Møller operators: $$\Omega^\pm(H,H_0):=\mathrm{s-lim}_{\tau\to\pm\infty}e^{i\tau ...
2
votes
1answer
20 views

Prove there cannot be an inner product which turns $l^p$ into an inner product space?

For all $1\leq p < \infty, \mbox{ }p$ is not equal to 2, prove there cannot exist an inner product that turns $(X,\|\cdot \|_p)$ into an inner product space; that is, prove that there cannot be ...
1
vote
1answer
18 views

Nonnormal Operator: Empty Spectrum

Are there operators on Hilbert space having empty spectrum? (Surely, for Banach spaces they do exists.) Necessarily, they must be closed and can't be normal.
0
votes
1answer
34 views

about weak convergence in $L^2(0,T;H)$.

Exercise Suppose $H$ is Hilbert space and $u_k$ converges weakly to $u$ in $L^2(0,T;H)$. Suppose further we have the uniform bounds $\mathrm{esssup}_{0≤t≤T} ||u_k(t)||≤C$. Then ...
2
votes
1answer
32 views

How does one diagonalise an operator that has exponential elements?

I asked this question before on the Physics StackExchange, but as one commenter noted I might have more luck here. So the question is: What is the diagonal form of the (density) operator $\hat\rho$, ...
1
vote
1answer
50 views

A Question on Compact Operators on Hilbert Space

I read this question which I have no idea how to start. Could anyone provide me with some detailed answer, please? Thanks. Suppose that a linear operator $F$ from a Hilbert space $\mathcal H$ to ...
1
vote
1answer
22 views

Hilbert transform on $L^p(\mathbb{T})$

Let $\infty >p\geq 2$, then for $f\in L^p(\mathbb{T})$ (here $\mathbb{T}=[0,1)$), show that for any real-valued trigonometric polynomial $f$, we have $H(f^2-(Hf)^2)=2fHf$. The hint is to use the ...