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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.

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$\newcommand{\R}{\mathbb{R}}$ $\DeclareMathOperator{\law}{Law}$ $\def\E{\hskip.15ex\mathsf{E}\hskip.10ex}$ Let $H$ be a Hilbert space equipped with the orthonormal basis $(e_i)_{i\ge1}$. Let $\xi=\...
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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 \...
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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 $r$ ...
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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 ...
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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 $W_\text{loc}^{\beta,2}(\cal X)$ (...
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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 ...
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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 ...
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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 ...
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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
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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(...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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,...
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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 ...
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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 ...
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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$$ ...
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Let $\ell^2(\mathbb{R})$ be the Hilbert space of square-summable sequences of real numbers, and let $\ell^2(\mathbb{Q})$ denote the dense subspace of square-summable sequences of rational numbers. Let ...
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Let $A $ and $ B $ be two self-adjoint and positive operators on a Hilbert space, and let $ 0 < b' \leq 2 \leq b $ and $ q > 1 $. Then, there exist constants ( C, C' > 0 ) such that: $$ (A^{b'...
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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 ...
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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 ...
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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 $\kappa$ ...
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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 ...
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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: $$ ...
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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 ...
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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^...
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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 ...
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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(...
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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 ...
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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 ...
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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 (...
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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 $...
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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 ...
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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 ...
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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$,...
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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 ...
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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 ...
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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
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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 ...
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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 $T$...
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I've often read that entanglement entropy in quantum field theory is ill-defined because local algebras are generally of type III, which implies that a trace doesn't exist. For a normal state $\omega_{...
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Consider the following regression problem $v=A(u) + \varepsilon$ for some operator $A:\mathcal{U} \rightarrow \mathcal{V}$ and some function spaces $\mathcal{U},\mathcal{V}$, mapping from $\mathcal{X}$...
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