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

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3
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1answer
42 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 ...
3
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2answers
181 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
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1answer
96 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
29 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?
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0answers
53 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
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1answer
95 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
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2answers
41 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
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1answer
118 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
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1answer
49 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
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1answer
21 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 ...
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0answers
54 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 ...
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0answers
107 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 $\sup$ ...
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1answer
37 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
179 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 ...
4
votes
1answer
209 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
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1answer
70 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 ...
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2answers
44 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 ...
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1answer
58 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
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1answer
19 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 ...
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1answer
105 views

Partial Isometries: Introduction

Attention This question has been modified drastically. It is done so the answer below is still correct. It is done so to allow more specialized threads. Problem How do I deal with partial ...
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0answers
33 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
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1answer
25 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
77 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 ...
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1answer
54 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 $x^2$ ...
3
votes
2answers
187 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 ...
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1answer
51 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 ...
3
votes
1answer
109 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 $C^\infty$ ...
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0answers
47 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
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1answer
106 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
88 views

Wave Operators: Hamiltonian

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 wave ...
0
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1answer
56 views

Wave Operators: Preliminary [closed]

Given a Hilbert space $\mathcal{H}$. Consider a free Hamiltonian $H_0$ and a perturbed one $H$. Introduce the wave operators: $$\Omega^\pm(H,H_0):=\mathrm{s-lim}_{\tau\to\pm\infty}e^{i\tau ...
2
votes
1answer
60 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 ...
2
votes
1answer
73 views

Closed Operators: Spectrum

Given a Hilbert space $\mathcal{H}$. Consider operators: $$T:\mathcal{D}(T)\to\mathcal{H}$$ Suppose one has: $$T=\overline{T}=T^{**}$$ Then it may happen: $$\sigma(T)=\varnothing,\mathbb{C}$$ What ...
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2answers
113 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
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1answer
46 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
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1answer
72 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
50 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 ...
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1answer
111 views

Inner products of weakly convergent sequences

I have a weakly convergent sequence in $L^2(U)$ (for $U$ some bounded open domain with smooth boundary), $u_k\rightharpoonup u$. I want to show that there is a sequence $v_k\rightharpoonup v$, such ...
0
votes
2answers
68 views

Spectral Measures: Numerical Range

Given a Hilbert space $\mathcal{H}$. Consider a normal operator $N:\mathcal{D}(N)\to\mathcal{H}$. The goal here is to prove: $$\langle\sigma(N)\rangle=\mathcal{W}(N)$$ By a previous result one has: ...
0
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1answer
67 views

An inequality about operator norm

Let $H$ be a Hilbert space and $T\in B(H)$, with $T_{i}\rightarrow T$ in strong operator topology. Then can we prove that $\liminf_{i\rightarrow \infty}||T_{i}||\geq ||T||$ ?
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1answer
57 views

A sum of a closed subspace and a closed one-dimensional space is closed

I'm losing my mind over this question. For $H$ a Hilbert space, $A,B$ closed subspaces, and $B$ is of dimension $1$, I want to prove that $A+B$ is also closed. I'm looking for a straightforward ...
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1answer
38 views

How can I calculate the projection of a Hilbert space into a closed subspace?

I was woundering if there is an easy way to calculate the projection of a Hilbert space into a closed subspace. Obviously one could write $P:H->C$ that is given by $P(x)=d$ s.a $d=inf||x-v||$ for ...
5
votes
1answer
225 views

Can the composite of two projections really fail to be a projection?

Let $H$ denote a Hilbert space. For any closed subspace $C \subseteq H$, write $P_C$ for the orthogonal projection onto $C$. Then according to wikipedia, the composite $P_U \circ P_V$ needn't be a ...
0
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1answer
59 views

$L_2$ as a Hilbert space and $\ell_2$

I know that, if measure $\mu$, with which measure space $X$ is endowed, has a countable base, i.e. if for any measurable $M\subset X$ there exists a measurable set $A_k\in\mathscr{A}$, where ...
0
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1answer
55 views

Example of a subspace S of a Hilbert space such that S^(⊥⊥) does not equal S?

I try to find an example of a subspace S of a Hilbert space H such that S^(⊥⊥) does not equal S. I know that subspace cannot be closed as for closed subspaces S^(⊥⊥)=S holds true. Does there exist ...
2
votes
1answer
97 views

Proving that if $<Ax,x>=0$ for every $x$, then $A$ is the zero operator

I feel kind of dumb but I needed this little claim as a part of a proof I'm writing, and I figured out that I'd better just ask, since I could not find the correct algebraic manipulation needed in ...
3
votes
1answer
48 views

Prove $Tx=(r_1x_1, r_2x_2, r_3x_3,…)$ is compact, $T:l^2\to l^2$, $r\in l^2$

Here is my question: Fix $r=(r_1,r_2,...)\in l^2$. Define $T:l^2\to l^2$ by $$Tx=(r_1x_1, r_2x_2, r_3x_3,...)$$ Prove that $T$ is compact. Here is what I have, input would be appreciated: Let ...
1
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1answer
52 views

Dense convex proper subset of the Hilbert space $l_2$: $\{x|\sum x_i=0\}$ [duplicate]

Let's consider the space $l_2$ (all sequences $x$ with $\sum x_i^2 < +\infty$) and its subset $Z = \{x|\sum x_i = 0\}$. I want to prove that the closure of $Z$ is $l_2$, but I can't. I tried to ...
3
votes
1answer
103 views

Definition of unitary operators

Let $\phi, \psi \in \mathcal{H}$ be some element from a hilbert space $\mathcal{H}$ and $U$ a linear operator $U: \mathcal{H} \rightarrow \mathcal{H}$. Does $$ \forall \phi: \| U \phi \|^2 = \| \phi ...
0
votes
1answer
82 views

Spectral Measures: Reducibility

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...