# Questions tagged [hilbert-spaces]

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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### Closure of the point spectrum of an unbounded diagonalizable operator

Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to$H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...
1 vote
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### What is lost after RKHS embedding of the L1 space?

We know that every distribution or $L^1$ function $f$ over space $\mathcal{X}$ (e.g., $R^d$) can be embedded to an RKHS $\mathcal{H}$ with a $1$-bounded kernel $\mathcal{K}$ (e.g., the RBF kernel) ...
1 vote
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### Properties of Sobolev spaces $W^{k,p}(\Omega, E)$ where $E$ is a Banach space

$\newcommand{\R}{\mathbb R}$Let $E$ be a Banach space with norm $\|\cdot\|_E$ and let $\Omega\subset \R^n$ be an open set. For $k\geq 0, p\geq 1$ we define $W^{k,p}(\Omega, E)$, the Sobolev space of ...
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### $f\in L^2_k(\mathbb R\times \mathbb S^1)$ implies that $t\mapsto f(t) \in L^2_a(\mathbb R, L^2_{k-a}(\mathbb S^1))$?

$\newcommand{\SS}{\mathbb{S}^1}$ $\newcommand{\R}{\mathbb{R}}$ Consider a function $f:\R\times \SS\to \R$ and suppose that $f$ is in the Sobolev space $L^2_k(\R\times\SS)$ for $k>1$ so that we can ...
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### Infinite-dimensional "algebraic varieties"

This question was formerly posted on MSE but did not receive any answer or comment, so I'm re-asking it here. Let $H$ be the infinite-dimensional seperable complex Hilbert space, and $P(H)$ denote its ...
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### A question about Marino–Prodi perturbation

In this paper N. Ghoussoub, the author claims the following version of Marino–Prodi perturbation, that is : Let $H$ a Hilbert space. Let $f\in C^2(H, \mathbb{R}),$ $K$ is a compact subset of $K_c$ (...
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### Convergence of solutions of regularised least square problems

Let $\mathcal{H}_1$ and $\mathcal{H}_2$ be two Hilbert spaces ($\mathcal{H_2}$ being finite dimensional, but I don't think that it matters). Consider $(A_{\lambda})_{\lambda>0}$ a familly of ...
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### Unicellular compact operators

An operator $T$ on a separable Hilbert space $H$ is called unicellular if any two closed invariant subspaces $M$ and $N$ are comparable; that is either $M\subseteq N$ or $N\subseteq M$. There are many ...
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### Concentration inequality for Hilbert space valued random variables

I have read in a paper about the following result: Let $V$ be a separable Hilbert space and $(\Omega,A_{\Omega},P)$ a probability space. Suppose that $Y_1,Y_2,...$ is a sequence of independent $V$-...
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### Is Solèr’s theorem true in constructive mathematics?

Solèr’s theorem says that for every star division ring $R$ and every $R$-module $H$ with an orthomodular Hermitian form $\langle (-),(-) \rangle:H \times H \to R$ such that there exists an infinite ... 64 views

### Mathematical reason for scatter states being special?

In infinite spectral theory, we have the discrete and continuous spectrum, which are called "bound" and "scatter states" in physics. My understanding is, if $O \in B(H)$ is a self-...
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I posted this on MSE a couple months ago and it got three upvotes but no answers or even comments so I decided to cross-post it here: For every pair $a,b$ of real numbers define the operator $U_{a,... 1 vote 1 answer 65 views ### Conditional Gaussians in infinite dimensions I asked this over on cross validated, but thought it might also get an answer here: The law of the conditional Gaussian distribution (the mean and covariance) are frequently mentioned to extend to the ... 1 vote 0 answers 50 views ### Maximal orthonormal set with fixed distance to subspace I'm trying to prove an upper bound on the size of a set$S$of orthonormal vectors in a finite-dimensional Hilbert space$\mathcal{H}$whose elements all have the same distance to the unit sphere of a ... 0 votes 0 answers 52 views ### Extending an unbounded dense linear functional Let$H$be an infinite dimensional separable Hilbert space over$\mathbb{C}$Let$V \subset H$be a dense subspace of$H$Let$f : V \to \mathbb{C}$be a unbounded functional linear My question is: Is ... 1 vote 1 answer 112 views ### A question on the self-adjointness of an operator Given a Hilbert space (separable)$\mathcal{H}$with an orthonormal basis$\{e_i\}_{i=1}^{\infty}$, define an operator$T$with domain$\mathcal{D}(T)$equal to the span of$\{e_i\}$by$Te_i:=\... 259 views

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### Is the "space of physical quantities" a field of transcendence degree $6$ or $7$ over the rationals?

Excuse my naive question and please let me explain it: In everyday life we experience 3 spatial "dimensions" + time etc. Usually the 3 dimensions are represented by a coordinate system and ...
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### Projection onto level set of convex functional

Fix a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and let $F:L^2_{\mathbb{P}}(\mathcal{F})\rightarrow (-\infty,\infty]$ be bounded-blow, convex, lower semi-continuous, and not identically ...
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### Infinite-dimensional analogue of "positive-negative splitting implies non-degeneracy"

(This question is related to Splitting a space into positive and negative parts but different.) Given a finite-dimensional vector space $V$ over $\mathbb{R}$, what I call a "positive-negative ...
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### Reverse martingale convergence theorem in Banach spaces

In section 1.5 of a course given by Gilles Pisier, the author is claiming that in the excerpt below $\operatorname E[\varphi_i\mid\mathcal A_{-n}]\to\operatorname E[\varphi_i\mid\mathcal A_{-\infty}]$ ...
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### Axiomatizing projective Hilbert spaces

This question arises in connection to trying to take a different (more intrinsic) perspective on the foundations of quantum mechanics, in which projective Hilbert spaces naturally arise, e.g. see ...
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### Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$

Remark: I cross-posted this question on MSE and added a bounty to it. Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly ...
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### Does the eigenvectors of a self-adjoint operator whose spectrum consists of simple eigenvalues construct a Hilbert basis?

I already asked this question on math.stackexchange a few days ago, but I still haven't received an answer, so I'm asking it here. Some PDE's have what is called a Lax pair i.e. there exists two ...
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### Geometry in Hilbert spaces / spheres in high dimensions

Let $H$ be a separable Hilbert space of infinite dimension and let $(e_n)_{n \in \mathbb{N}}$ be an orthonormal basis of $H$. For a series $(\alpha_n)_{n \in \mathbb{N}} \subset \mathbb{R^+}$ we are ...
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### Proof: If a reproducing kernel exists for a Hilbert space, then it is unique

I really want to prove the statement in the title but I'm struggling with it. Here my current state: Proof via contradiction. Let $\mathcal{H}$ be a RKHS with two reproducing kernels $k$ and $\hat{k}$ ...
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### Infinite dimensional topological quantum field theories?

A topological quantum field theory is a functor $Z:Cob(n) \to Hilb$ mapping from the $n$-dimensional cobordisms to $Hilb$, the category of Hilbert spaces with morphisms being bounded linear operators. ...
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### Criterion for compactness

Let $T:H\to H$ be a continuous operator on a Hilbert space. Assume there exists an orthonormal base $(e_j)_{j\in\mathbb N}$, such that the sequence $Te_j$ tends to zero. Must $T$ be compact? 67 views

### When does an RKHS contain another?

Consider a psd kernel function on the unit-sphere in $\mathbb R^d$ off the form $K(x,x') = \varphi(x^\top x')$ for some $\varphi:[-1,1] \to \mathbb R$, and let $\mathcal H_\varphi$ be the induced ...
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### Drinfeld center of a tensor category

Firstly, apologies for the imprecise language, I'm a physicist trying to understand anyonic excitations from the lens of category theory. If I have a category (say $\operatorname{Rep}(\mathbb{Z}_2)$) ...
I have discovered a method to measure the similarity of two successive musical notes which I wanted to share with a question: It is known in music theory that two successive pitches $a,b$ which sound “...