Skip to main content
MathOverflow

Questions tagged [hilbert-spaces]

Ask Question

A Hilbert space $H$ is a real or complex vector space endowed with an inner product such that $H$ is a complete metric space when endowed with the norm induced by this inner product.

766 questions
Newest Active Bountied Unanswered
Filter by
Sorted by
Tagged with
4 votes
1 answer
295 views

Let $\{H_i\}_{i\in I}$ be a directed system of Hilbert spaces over a poset $I$ such that $H_i\subset H_j$ whenever $i\leq j$. We consider the direct limit $H=\bigcup_{i\in I} H_i$ and equip it with ...
Knowvice's user avatar
  • 63
1 vote
0 answers
57 views

Let $J_0$ be a (separable) complex Hilbert space with scalar product $\langle \cdot,\cdot\rangle_{J_0}$ and norm $\|\cdot\|_{J_0}$. Suppose we are given a real, symmetric, non-negative bilinear form $$...
N_Nehmer's user avatar
  • 11
1 vote
1 answer
148 views

$\newcommand{\R}{\mathbb{R}}$ $\DeclareMathOperator{\law}{Law}$ $\def\E{\hskip.15ex\mathsf{E}\hskip.10ex}$ Let $H$ be a Hilbert space equipped with the orthonormal basis $(e_i)_{i\ge1}$. Let $\xi=\...
Oleg's user avatar
  • 961
0 votes
0 answers
92 views

Suppose $X$ is a (separable) Banach space and $T\in\mathcal{B}(X)$ a bounded operator. Let $\mathcal{A}$ be the unital algebra generated by the resolvent operators $\{(\lambda I- T)^{-1}: \lambda\in \...
Markus's user avatar
  • 1,361
4 votes
1 answer
172 views

Let $E$ be a Banach space. Let $B$ be the closed unit ball of $E$ endowed with the restriction of the weak topology of $E$. For $e\in E$, $r\in\mathbb{R}$ let $B(e,r)$ be the closed ball of radius $r$ ...
erz's user avatar
  • 5,693
3 votes
0 answers
170 views

I plan on giving a $75$ minutes talk on a general subject of my choice. The audience will comprise undergrads (with knowledge of calculus, linear algebra, probability, groups and probably some Hilbert ...
GBA's user avatar
  • 277
3 votes
0 answers
150 views

Let ${\cal F}_k$ be the RKHS of functions on an open set $\cal X\subseteq {\bf R}^n$ with kernel $k$. For which $k$ can ${\cal F}_k$ be embedded in the Sobolev space $W_\text{loc}^{\beta,2}(\cal X)$ (...
Wicher's user avatar
  • 93
7 votes
1 answer
434 views

I have a quick question regarding the spectral calculus of unbounded operators. Let $(\mathcal{H},\langle\cdot,\cdot\rangle_{\mathcal{H}})$ be a Hilbert space and $A:\mathcal{D}(A)\to\mathcal{H}$ be ...
Dheghom's user avatar
  • 326
2 votes
1 answer
300 views

I am studying the problem of representing positive linear functionals on the space of bounded norm-continuous functions on a separable infinite-dimensional Hilbert space $H$. A known challenge is that ...
Zlyp's user avatar
  • 341
1 vote
0 answers
106 views

Let $\mathcal{H}$ be a complex (finite dimensional) Hilbert space. The norm $$\mathcal{H}-\{0\}\to\mathbb{R},\,x\mapsto \|x\|$$ is a Kahler potential for a Riemannian metric. Explicitly, this metric ...
Josh L's user avatar
  • 11
5 votes
0 answers
156 views

Let $H$ be a separable infinite-dimensional Hilbert space over $\mathbb{R}$ with norm $\|\cdot\|$. Let $\{\mu_\alpha\}$ be a net of Borel probability measures on $H$, and let $\mu$ be a Borel ...
Zlyp's user avatar
  • 341
6 votes
1 answer
209 views

Let $H$ be a separable infinite-dimensional Hilbert space over $\mathbb{R}$ with inner product $\langle \cdot, \cdot \rangle$ and norm $|\cdot|$. Define the function $f\colon H \to \mathbb{R}$ by $$ f(...
Zlyp's user avatar
  • 341
0 votes
1 answer
69 views

All references refer to "Finite Element Methods for Maxwell's Equations" by Monk. Preliminaries: Let $\Omega\subset \mathbb{R}^3$ be a bounded lipschitz domain. The space $H(\text{curl})$ is ...
Mandelbrot's user avatar
  • 135
6 votes
1 answer
462 views

The following is a repeat from the following question on MathStack Exchange. I am just hoping to have more success here. This is a follow-up from that question. The question is this: I want to ...
Gateau au fromage's user avatar
  • 555
3 votes
0 answers
125 views

Let $X$ be a subset of the Lebesgue space $L_2([0,1])$ satisfying the following: For every $x \in X$ we have $0\le x\le \mathbf{1}$. $X$ is norm compact as a subset of $L_\infty([0,1])$. This means ...
Rabee Tourky's user avatar
  • 930
1 vote
0 answers
314 views

I have posted this question on MathSE, but posting it here since I have not received any answers there. I am trying to understand the relation between the spectrum of a multiplication operator on a ...
Mutasim Mim's user avatar
  • 153
3 votes
1 answer
232 views

If we are talking about the Euclidean space $\mathbb{R}^n$, then we may naturally measure what part of the whole space does the Borel set $A$ occupy by simply introducing the notion of the upper ...
Oleksandr Liubimov's user avatar
  • 778
21 votes
1 answer
1k views

The following problem I knew for $5+$ years as a medium-easy exercise in basic functional analysis, until last year when I planned to give it to my students, and to my horror realized that the ...
Aleksei Kulikov's user avatar
  • 8,306
0 votes
0 answers
85 views

I am searching for rational embeddings of positive definite kernels $k$ symmetric, taking rational values and $k(n,n)=1$ such as $k(a,b) = \min(a,b)/\max(a,b)$ or $k(a,b) = 2\gcd(a,b)/(a+b)$ or $k(a,...
mathoverflowUser's user avatar
  • 1
0 votes
0 answers
64 views

I'm investigating a particular topic and I'd like to get some references on it. The idea is as follows: pick some natural $d$ and let $\mathcal{F}_d$ be a Gaussian Process on $\mathbb{R}^d$ with mean ...
Daniel Goc's user avatar
  • 103
0 votes
1 answer
116 views

Let $H$ be a separable Hilbert space and consider the map sending any Fréchet differentiable non-linear functional $F\in C^1(X)$ and let $\nabla F\in L(X)$ denote its Fréchet gradient. Consider the ...
user558721's user avatar
  • 57
2 votes
0 answers
97 views

I have two sets of countably many vectors $\{ u_k \}$, $\{ v_k \}$ lying on a finite-dimensional space $\mathbb{C}^K$. They satisfy the relation $$\langle u_k, u_h \rangle = \langle v_k, v_h \rangle$$ ...
NYG's user avatar
  • 149
5 votes
0 answers
177 views

Let $\ell^2(\mathbb{R})$ be the Hilbert space of square-summable sequences of real numbers, and let $\ell^2(\mathbb{Q})$ denote the dense subspace of square-summable sequences of rational numbers. Let ...
Daniel Asimov's user avatar
  • 3,557
0 votes
0 answers
110 views

Let $A $ and $ B $ be two self-adjoint and positive operators on a Hilbert space, and let $ 0 < b' \leq 2 \leq b $ and $ q > 1 $. Then, there exist constants ( C, C' > 0 ) such that: $$ (A^{b'...
zoran Vicovic's user avatar
  • 571
3 votes
1 answer
240 views

Hilbert bundles are coming up in my research, and I’m trying to better understand them. Since there are multiple definitions of Hilbert bundles, I will clarify that I’m working with the definition in ...
Joe's user avatar
  • 195
9 votes
0 answers
337 views

Is there a characterization of the objects of the category of Banach spaces (with either contractions or bounded linear maps as morphisms, although I would expect the former to be easier) that are ...
Cameron Zwarich's user avatar
  • 4,626
9 votes
1 answer
717 views

For each $p \in \mathbb{R}$ and each cardinal $\kappa$, write $\text{L}_{\text{p}}^{\kappa}$ for the $\text{L}_{\text{p}}$-space one obtains from completing a real vector space of cardinality $\kappa$ ...
user avatar
5 votes
1 answer
310 views

I read in a post here wherein it is claimed that given a perfect measure space $(\Omega, \mathcal F, \mu)$, a Banach space $Y$, and a function $f: \Omega \to Y$ that $f$ is Bochner measurable if and ...
Trajan Murphy's user avatar
  • 51
1 vote
1 answer
138 views

Let $\mu$ be a centered Gaussian measure on a separable infinite-dimensional Hilbert space $H$. For every $\delta>0$ does there exist a convex and compact set $C_{\delta}\subseteq H$ such that: $$ ...
Mathematical-Semi_N00b's user avatar
  • 476
10 votes
1 answer
648 views

Consider the unitary operator $U$ on the Hilbert space $H:=L^2([0,\frac\pi2])$, that takes $\cos((2k+1)x)$ to $\sin((2k+1)x)$, for $k\in\mathbb N$ (both are orthogonal basis). How can we explicitly ...
Pietro Majer's user avatar
  • 63.5k
3 votes
0 answers
252 views

Consider the Hilbert space $\ell^2(\mathbb{Z})$ and unbounded operators $u$ and $v$ defined as $u: f_k \to f_{k+1}$ and $v: f_k \to q^k f_k$ for some fixed $0<q<1$. Together with $u^{-1}$ and $v^...
Davide's user avatar
  • 31
7 votes
1 answer
619 views

If $G$ is a locally compact group, the definition of unitary dual of $G$ is the set of all equivalence classes of irreducible unitary representations of $G$, denoted by $\widehat{G}$. But why this is ...
jack's user avatar
  • 99
3 votes
0 answers
124 views

In Hilbert space modeled by $L^2([0,1])$ we can define a set $B^+=\{x\in B(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ and $S^+=\{x\in S(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ where where $B(...
Józef Zápařka's user avatar
  • 31
5 votes
1 answer
314 views

Let $H$ be a separable Hilbert space. Let $L_1$ denote the space of trace class operators on $H$ with the trace-class norm $\|\cdot\|_1$, i.e. $\|K\|_1=Tr|K|$ for all $K\in L_1$. Are there easy ...
lulli_'s user avatar
  • 59
3 votes
0 answers
127 views

Consider two Hilbert spaces $H_1$ and $H_2$, and $A$, $B$ unbounded operators on $H_1$, $H_2$ respectively. $(A \otimes I)$ is classically defined as the closure of the operator defined on the set of ...
Hugo's user avatar
  • 31
0 votes
0 answers
109 views

At a very high level, I’m interested in the following question. Suppose $X$ is a (separable) Hilbert space, and $T_n : X \rightarrow X$ is a sequence of finite rank self-adjoint maps that converges (...
Joe's user avatar
  • 195
1 vote
0 answers
154 views

Let $G$ be a compact Lie group with a biinvariant metric. Note that $G\times G$ acts isometrically on $G$ from left and right. Consider the quotient $D=G/H$ by a closed subgroup $H\le G\times G$; if $...
Anton Petrunin's user avatar
  • 47.1k
5 votes
2 answers
275 views

I am linearizing nonlinear waves and get operators of the form below. Everything is considered in $L^2(\mathbb{R})$. Consider the operator $L_1=\frac{d}{dx}$. The domain is $H^1(\mathbb{R})$ and it is ...
Gateau au fromage's user avatar
  • 555
2 votes
0 answers
125 views

I had a hard time deciding whether this question is more appropriate for physicsSE or MO, so I have cross-listed it for the time being. The physicsSE post can be found here. In the lecture Is ...
CBBAM's user avatar
  • 883
6 votes
1 answer
248 views

I’m currently studying orthomodular lattices arising out of operator algebras. One of the most standard examples is the projection lattice of a von Neumann algebra - if $M$ acts on a Hilbert space $H$,...
David Gao's user avatar
  • 5,085
1 vote
0 answers
76 views

This is related to a question I recently asked on math.SE. Consider a subset $G\equiv \{g_k\}_{k\in\mathbb{N} }\subseteq\mathcal H$ in a separable Hilbert space $\mathcal H$, and suppose $G$ spans the ...
glS's user avatar
  • 352
2 votes
1 answer
247 views

Let $X$ be a locally compact space, let $H$ be a Hilbert space and let $\beta:X\to H$ be a continuous function such that the linear subspace of $H$ spanned by $\beta(X)$ is dense in $H$. I would like ...
P. P. Tuong's user avatar
  • 529
-1 votes
1 answer
135 views

Let the following be hilbert spaces with dens inclusions $V ↪H=H^* ↪V^*$. Where $H^*$ and $V^*$ are the duals. $H$ has the product $(*,*)$ and $V×V^*$ has the product $⟨*,*⟩$. Let $u∈L^2 ([0,T];V); ...
Alucard-o Ming's user avatar
  • 181
7 votes
1 answer
2k views

Recently four mathematicians claimed to have proven the invariant subspace problem, which is the problem that states Does every bounded operator on a separable Hilbert space have a non-trivial ...
euleroid's user avatar
  • 97
1 vote
0 answers
138 views

I have a family of operators $T_n\colon X \to Y$ where $X,Y$ are Hilbert spaces. These operators are nonlinear. What kind of notions of convergence does one have for such operators? I'm specifically ...
C_Al's user avatar
  • 261
1 vote
2 answers
336 views

Let $A\subset \ell^2$ consist of all $x\in \ell^2$ with $|x|_2=1$ which does not belong to any $\ell^p$ for all $0<p<2$. Note that $A$ is non-empty with a Baire category argument. I ...
Ali Taghavi's user avatar
  • 400
0 votes
0 answers
85 views

I’m interested in using laplacian (−Δ) eigenfunction as a basis for H1(Rn) . I know that in H1(Ω) , Ω bounded this can be done so I was wandering about H1(Rn) . Now let eλ be an eigenfunction ...
Alucard-o Ming's user avatar
  • 181
6 votes
0 answers
256 views

Let $\mathcal{S}'(\mathbb{R}^d)$ be the set of tempered distributions over $\mathbb{R}^d$ and $d\phi_C$ a Gaussian measure over $\mathcal{S}'(\mathbb{R}^d)$ with covariance operator $C$. Consider the ...
CBBAM's user avatar
  • 883
5 votes
1 answer
255 views

Let $H_n$ denote the space of $n\times n$ Hermitian matrices. For every $A\in H_n$, using the spectral decompostion of $A$, $$A=\sum_i \lambda_i x_ix_i^*,$$ one can define the positive and negative ...
Mostafa - Free Palestine's user avatar
  • 4,503
10 votes
0 answers
270 views

I'm cross-posting this question from Math.SE, as it didn't get much attention there. Let $H$ be a separable Hilbert space and $T \in L(H)$ a trace-class operator. It is well known that the trace of $T$...
WillG's user avatar
  • 235

1
2 3 4 5
…
16