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

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2
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1answer
86 views

Reiterate Volterra integral operator is a contraction

I read in Kolmogorov and Fomin's Элементы теории функций и функционального анализа (p. 472 here) the statement that Volterra operator $A:L_2[a,b]\to L_2[a,b]$ defined ...
1
vote
1answer
33 views

Calculation of operator norm

$H$ is a Hilbert space, $T: H \to H$ linear bounded operator, $||T||$ is the norm of $T$ given by $$||T||=\sup\{||T(x)||;||x||\le 1 \}. $$ Is it true that $$||T||=\sup\{|\langle ...
0
votes
1answer
33 views

$\ker (I-A)=\{0\}\Rightarrow\text{im }(I-A)=H$ for $A:H\to H$ compact

Let $T$ be the operator defined by $T:=I-A$ where $I:H\to H$ is the identity and $A:H\to H$ is a compact operator defined on Hilbert space $H$. In such a case, if we defined the chain of ...
1
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0answers
40 views

the spectral radius of normal operator

Let $H$ be a Hilbert space and $T$ be linear bounded operator in $H$. Prove that if $T$ is normal then the spectral radius of $T$, $$r(T)=\|T\|.$$ Is this TRUE?
1
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1answer
71 views

Direct sum of kernel and image of the adjoint operator

Let $H$ be a separable Hilbert space and $T:=I-A$, where $A:H\to H$ is a compact operator. If $T^\ast$ is the adjoint operator of $T$ it can be proved that $\ker T$ and $\text{im } T^\ast:=T^\ast (H)$ ...
1
vote
1answer
24 views

Degenerate Hilbert-Schmidt operators

Let us define a Hilbert Schmidt operator $A:L_2[a,b]\to L_2[a,b]$ by $$A\varphi:=\int_{[a,b]} K(s,t)\varphi(t)d\mu_t$$where $\mu_t$ is the linear Lebesgue measure. A degenerate case is represented by ...
0
votes
3answers
76 views

Need countereample : If a sequence $(a_n) \in l^2 $ , then the sequence $(1/a_n) \notin l^2 $

I want to know the counterexample for the following statement : Given a sequence $(a_n)$ such that $a_i\ne 0 $ for any $i$ : If the sequence $(a_n) \in l^2 $ , then the sequence $(1/a_n) \notin l^2$. ...
3
votes
0answers
23 views

Limits of trajectory of gradient flow in Hilbert space

I have been studying about gradient flow in Hilbert space of a Morse function $f$. Specifically, let $X$ be a Hilbert space and $f : X\to \mathbb R$ be $C^3$ function. The gradient flow here is ...
0
votes
1answer
26 views

CAR-Algebra: Nontriviality?

Given a Hilbert space $\mathcal{h}$. Consider the abstract CAR-algebra $a:\mathcal{h}\to\mathcal{A}_\text{CAR}$. Then their actually isometries: $$a:=a(f):\quad ...
2
votes
1answer
33 views

How to show that a vector space is closed?

I am trying to complete a proof which requires me to prove that a subspace $H$ of $L^2{(\Omega,\mathcal{F}},\mathbb{P})$ is closed vector space in $L^2{(\Omega,\mathcal{F}},\mathbb{P})$ What do I ...
2
votes
2answers
140 views

Dense subspaces, closed subspaces and unbounded operators in Hilbert spaces

Let $\mathcal{H}$ be a Hilbert space, and let $N\subseteq\mathcal{H}$. I found two interesting statements (without proof): if a closed subspace $N$ is such that $N^{\perp}=\{0\}$ (which is ...
0
votes
1answer
39 views

$H_0^1(\Omega)$ with $(u,v)_{H_0^1} = \int\nabla u \cdot \nabla v$

When $\Omega$ is a bounded open set of $\mathbb{R}^N$ with the help of Poincare inequality, we know that $H_0^1(\Omega)$ with $(u,v)_{H_0^1} = \int\nabla u \cdot \nabla v$ is a Hilbert space. ...
0
votes
1answer
12 views

If $H$ is a Hilbert space and $T$ an isometric operator, then $\overline{R(T-I)}=H \implies N(T-I)=\{0\}$?

Let $H$ be a Hilbert space. Let $T$ be a linear operator and $R(T)$, $D(T)$, $N(T)$ the range, domain and kernel of $T$, respectively. If $\|Tx\|=\|x\|$ for all $x \in D(T)$, then $T$ is called an ...
0
votes
0answers
45 views

Subdifferential of a continuous function is non-empty

Prove that if $X$ $-$ normed vector space, $x_0 \in \text{int }A$ and convex function $f$ is continuous in $x_0$ then $\partial f(x_0) \neq \emptyset$. $\partial f(x_0) = \{x^*\in X^*:\forall x \in ...
3
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2answers
186 views

Tensor Product: Hilbert Spaces

This question has been modified... Problem Given Hilbert spaces. In general, their algebraic tensor product isn't complete: ...
1
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2answers
31 views

Which Sobolev-Space to use to formulate weak biharmonic equation, $H^2_0$ or $H_0^1\cap H^2$?

For the weak formulation of the biharmonic equation on a smooth domain $\Omega$ $$ \Delta^2u=0\;\text{in}\;\Omega\\ u=0, \nabla u\cdot \nu=0\; \text{on}\; \partial\Omega $$ why does one take ...
0
votes
0answers
27 views

Characterization of Hilbert spaces among Banach spaces

Let $H$ be a complex Hilbert space. I know that $H\simeq \overline{H^*}$ by the Riesz representation theorem, where $\overline{X}$ means the complex conjugate space of $X$. I want to prove the ...
1
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0answers
55 views

Dichotomy for global existence or blow up for solutions of evolution problems.

Consider the problem (Nonlinear Schrödinger equation) \begin{equation} \left\{ \begin{array}{rl} iu_t + \Delta u\mp u|u|^{\alpha}=0\\ u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\ ...
0
votes
0answers
27 views

Compact operator space Hilbert

Let $H_1$ and $ H_2$ Hilbert space and $T:H_1\rightarrow{H_2}$ a compact operator. Shows $N(T)^\perp \subseteq H_1$ is a subspace separable of $H_1$. indeed as $N(T)^\perp$ is a closed subspace of ...
1
vote
1answer
155 views

double Orthogonal complement is equal to topological closure

So I'm in an advanced Linear Algebra class and we just moved into Hilbert spaces and such, and I'm struggling with this question. Let $A$ be a nonempty subset of a Hilbert space $H$. Denote by ...
1
vote
1answer
30 views

Does Convergence of Maps Evaluated at Points Imply Convergence in Operator Norm?

Suppose that I have $T,T_n \in B_H$, for some Hilbert space $H$. Is the following implication true? $$ \|(T-T_n)x\| \rightarrow 0 \ \forall x\in H \ \Rightarrow \ \|T-T_n\| \rightarrow 0, \ \text{ie} ...
4
votes
1answer
82 views

Uncountable series without axiom of choice

Consider a sequence of positive real numbers $(\alpha_i)_{i\in I}$ for some (suppose maybe wellordered for now) set $I$. Using axiom of choice, it is easy to see that $\sum_i \alpha_i$ is infinite if ...
4
votes
1answer
141 views

Brownian motion, reproducing kernel Hilbert space, and the Laplace operator

Consider the standard Brownian motion on $[0,1]$: $$ dB_t, \; B_0 = 0, $$ defined on the probability space $(\Omega, P)$. It covariance function is $K(s,t) = \min \{s , t\}$ on $[0,1] \times ...
2
votes
1answer
66 views

Reproducing kernel Hilbert space, why?

Let $K: X \times X \rightarrow \mathbb{C}$ be a positive definite kernel on a set $X$, i.e. for any $x_1, \cdots, x_n \in X$, the matrix $$ [K(x_i, x_j)]_{ij} \in \mathbb{C}^{n \times n} $$ is ...
1
vote
1answer
60 views

Weak convergence in Hilbert space implies strong convergence of averages for some subsequence

Let $H$ Hilbert Space. Show that if $x_n\rightharpoonup x$ then there exists a subsequence $\{x_{nk}\}$ of $\{x_{n}\}$ such that the sequence $\lim_{m\rightarrow \infty } ...
1
vote
1answer
106 views

Hermite functions as eigenvectors of Fourier transform

In order to find an orthogonal basis of eigenvectors of the Fourier transform operator $F:L_2(\mathbb{R})\to L_2(\mathbb{R})$, $f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x$ for ...
0
votes
1answer
52 views

Proving dense set is core for a self adjoint operator

Let $A$ be a self adjoint operator in a Hilbert space $H$ and $D\subseteq D(A)$ a dense subset such that $$ e^{iAt}:D \to D. $$ How can I show that $D$ is a core for $A$? I need to show that ...
1
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0answers
39 views

Green-Operator for Sturm-Liouville Differential equation compact on Sobolev space?

Let $g$ be Green's Function for a Sturm-Liouville differential equation. Is the operator $G: H_{0}^{1}(0,1) \rightarrow H_{0}^{1}(0,1)$ defined by $(Gf)(x) := \int_{0}^{1} g(x,y)f(y) dy, \quad f \in ...
3
votes
1answer
54 views

Is the intersection between two $n$-spheres an $(n-1)$-sphere?

It is true that the intersection between two $n$-sphere in $\mathbb{R}^n$ is a $(n-1)$-sphere if is not empty or a single point? I have tried to prove it but my only idea is to work with equations and ...
2
votes
2answers
82 views

Stone's Theorem Integral: Avanced Integral

Reference This problem grew out from: Stone's Theorem Integral: Basic Integral Problem Given the real line as measure space $\mathbb{R}$ and a Hilbert space $\mathcal{H}$. Consider a strongly ...
1
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0answers
14 views

The median in a isosceles triangle is ortoghonal into a hilbert space

how can I prove that if $p$, $q$, $r$ and $o$ are points in a Hilbert space such that $p$, $q$, $o$ are collinear, $\|p-o\|=\|q-o\|$ and $\|p-r\|=\|q-r\|$ then $r-o \perp p-o$?. I think it's a ...
3
votes
2answers
70 views

Book for Hilbert spaces.

Which book either on functional analysis or specifically for Hilbert spaces has the best way of explaining with most examples and to the point without much applications. I studied Limaye's book and ...
0
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1answer
30 views

Basis of $W^{1,p}_0\cap L^2$ using $(\lambda_i, v)_{H^s_0} = \mu_i(\lambda_i, v)_{L^2}$

Let $p > 1$. Define $\lambda_i$ by the eigenfunctions of the problem $$(\lambda_i, v)_{H^s_0} = \mu_i(\lambda_i, v)_{L^2}\quad\text{for all $v \in H^s_0(\Omega)$},$$ where $s$ is chosen ...
0
votes
1answer
54 views

Spectral Measures: Spectral Spaces (II)

Problem Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Denote its probability measures by: ...
0
votes
1answer
31 views

An example of an unbounded non-orthogonal projection in a Hilbert space

What is an example of an unbounded non-orthogonal projection in a Hilbert spaces? Does it exist? A non-orthogonal projection is an idempotent operator: $T^2=T$. So the question is: can such an ...
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0answers
86 views

Orthonormal set is a Hilbert basis $\iff$ Parseval's identity is true

Let $H$ be a Hilbert space and $\{e_k:k\in \mathbb{Z}\}$ an orthonormal set. Prove that the set is a Hilbert basis if and only if Parseval's identity is true. The direct theorem is almost ...
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0answers
46 views

Cardinality of dense subset of Hilbert space

If $H$ is an infinite dimensional Hilbert space, I want to show that any total orthonormal family in $H$ has the same cardinality as the minimum cardinality of a dense subset of $H$ but I am ...
7
votes
3answers
209 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
62 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
55 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
41 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 ...
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0answers
22 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
94 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 ...
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1answer
86 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
29 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
55 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
51 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
22 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
39 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
76 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 ...