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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Orthonormal bases on Reproducing Kernel Hilbert Spaces

Recall that a Hilbert space $\mathcal{H}$ is a reproducing kernel Hilbert space (RKHS) if the elements of $\mathcal{H}$ are functions on a certain set $X$ and for any $a\in X$, the linear functional ...
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42 views

Matrix representation of linearized PDE

My motivation for this question is to investigate the linear stability of steady solutions of a nonlinear PDE by computing eigenvalues of the linearized equations. Specifically, I have a function ...
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1answer
249 views

A Hilbert space characterization via retractions--a conjecture

Given a Banach space $X$ and a functional $f:X\rightarrow \mathbb R$, let $$ X_f := \{x\in X : f(x)\ge 0\} $$ ("functional" means "non-zero linear functional"). Also, given a topological space $E$ ...
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93 views

Gleason’s Theorem for non-separable Hilbert spaces: On a theorem of Solovay

Gleason’s theorem plays an important role in the foundations of quantum mechanics. On the positive side it demonstrates how the probabilistic structure of quantum theory follows from its logical ...
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18 views

Efficient packing in Gaussian measures on Hilbert spaces

Let $B(0,1)$ be the unit ball in a separable Hilbert space $H$ with a Gaussian measure $\mu$ on it. For a small $r > 0$, can we have $x_i \in B(0,1)$, $0 < r_i \leq r$ and a constant $K > 0$ ...
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80 views

orthogonal projector onto the set of convex functions

Let $\Omega\subset \mathbb R^d$ be an open, convex domain, and consider the Hilbert space $L^2(\Omega)$. Each sum of convex functions is convex, hence the subset $Conv(\Omega)$ of all convex functions ...
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1answer
94 views

Discrete Wavelets

I am looking for research that has been done in Discrete wavelets. Let me be specific as Google doesn't give me what I want when I say "discrete wavelets". I don't want countable basis for $ ...
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0answers
39 views

Hilbert Manifold structure on global sections of a sheaf

Let $<U_i,\phi_i>$ be an atlas on a Hilbert manifold $M$. If $F(M,U_i)$ is a Hilbert space for every $U_i$ then does this imply that $F(M,M)$ can be made into a Hilbert manifold also with ...
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2answers
158 views

Density of Gaussian measures on Banach spaces

I am trying to get my head around this question and was reading (1) which states the same a little bit more general: Let $X$ be a separable Banach space and $X^*$ the dual space. The mean value ...
4
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1answer
136 views

Is the sum of spectral projections a projection?

Let $T$ be a closed operator on a Hilbert space with discrete spectrum. Then for $\{\lambda_1,...\lambda_n\}\in\sigma(T)$ one can define the spectral projections ...
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1answer
65 views

Embedding Riemmanian Manifold Linearly

Given a Hilbert Manifold $M$ does there exist a smooth map into some very large Hilbert space taking geodesics to straight lines?
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Are dagger categories truly evil?

Recall that a dagger category is a category equipped with an involution $*:Hom(x,y)\to Hom(y,x)$ that satisfies $f^{**}=f$ and $f^* g^*=(gf)^*$. A prominent example of a dagger category is the ...
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64 views

Norm of derivative of rank one projector

I asked this question on math.stack but I got no answer, so I try here. Let $\phi(t)$ be a solution for the nonlinear Schroedinger equation\begin{equation} ...
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0answers
138 views

Deformation and Representations

Let $\widetilde{U_q(sl_n)}$ denote a deformation of the algebra $U_q(sl_n)$. In particular, $\widetilde{U_q(sl_n)}$ is defined by the same generators and relations and $*$-operations as $U_q(sl_n)$ ...
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51 views

Zeros of functions constituting a Riesz-basis for the Paley-Wiener space

I have a short question which first requires some slightly elaborate definitions: Let $(e_n)$ be a Riesz-basis for a Hilbert space $\mathcal{H}$ with biorthogonal basis $(g_n)$, i.e. $\langle e_m, ...
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2answers
57 views

Distribution of the RKHS norm of the posterior of a Gaussian process

In a classical noisy regression setting, let $\big(f(x)\big)_{x\in\cal X}$ be a centered Gaussian process of covariance $k$ on a compact $\cal X$, and $\mathcal{F}_n$ be the filtration generated by ...
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0answers
40 views

Equivariant exponential map on Hilbert manifolds

Let $M$ be a Hilbert Riemannian manifold and $G$ a finite-dimensional Lie group acting on $M$. It is well-known that when $M$, too, is finite-dimensional $$\exp_p: U \subset T_pM \rightarrow \exp_p(U) ...
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1answer
146 views

On the Riesz representation theorem II

I have a follow-up question to On the Riesz representation theorem . Let $V$ be a subspace of a Hilbert space, and let $V^\times$ be the space of all antilinear functionals on $V$, equipped with the ...
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1answer
349 views

On the Riesz representation theorem

Let $V$ be a vector space with inner product $(\phi,\psi)$ antilinear in the second argument - not necessarily a Hilbert space. Let $\Phi$ be an antilinear functional on $V$. What are the precise ...
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1answer
482 views

An equivalence relation on the space of polynomials in one complex variable

Let $P(z)$ be a polynomial with complex variable $z$. We consider the following distribution for the roots of $P(z)=0$: the distribution is a triple $(n_{1},n_{2},n_{3})$ where these integers are ...
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0answers
99 views

Algorithm for finding eigenfunctions

I have an $ L^2(\mathbb{R}) $ operator that looks like $$ \Omega = \int \partial\phi(a, b)\ \ |b, a\rangle \langle b, a |, $$ where $ \langle x | a, b \rangle = f_a(x - b) e^{x^2/2} $ and $ f_a \in ...
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1answer
176 views

No Hilbert space can have countable Hamel basis without using Baire's Category theorem [closed]

I want to prove that no Hilbert space can have countable Hamel basis just using the fact that any finite dimensional subspace is closed (more specifically without using Baire's theorem). I saw a paper ...
2
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1answer
159 views

Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that I could not find it any where. Is the completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm $$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, ...
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548 views

Schur's Lemma for Hilbert spaces

Let $H$ be a complex Hilbert space and let a group $G$ act on $H$ such that there are no invariant closed subspaces besides $H$ and $(0)$. Let $D$ be the ring of bounded operators which commute with ...
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1answer
67 views

Continuous section inside a family of rank-varying operators

Good morning everybody, my question is as follows: let $K$ be a compact set and assume $F:K\to L(\mathcal H,\mathbb R^{m+1})$ be a continuous map from the compact set $K$ to the space of linear ...
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149 views

How to change the given metric if we want to add few extra isometries?

I have a Hilbert Space $X$ and a group $G$, which consists of bounded linear self-bijections of $X$ (if it helps, this group has a locally compact, but not compact topology). Is there a canonical way ...
5
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1answer
160 views

Stinespring's dilation without $C^{\ast}$-algebras

Does Stinespring's dilation theorem hold if the algebra of interest is a topological $\ast$-algebra instead of the usual $C^{\ast}$-algebra? I will now state the version of Stinespring's dilation ...
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1answer
197 views

Is :$\frac{\Bbb d}{\Bbb d x}$ a chaotic operator in infinite-dimensional Hilbert space? [closed]

I proposed this question in SE but no answer ,may I have a problem in my question, I would like to know when $\frac{\Bbb d}{\Bbb d x}$ does chaotic operator in Hilbert space ? Let $H$=$L^2(\mathbb ...
3
votes
1answer
110 views

Sum of two parts of a continuous stochastic process

Let $X$ be a centered continuous stochastic process which is square integrable on $[0,2]\times \Omega$ and the basis of $L^2(0,2)$ is $\{e_i\}$. By using Karhunen-Leove Theorem one can write for all ...
5
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1answer
273 views

Is the unitary group of a pre Hilbert space contractible?

I already posted my question on mathstackexchange For a separable Hilbert space $H$ it is known that the unitary group $U(H)$ is contractible, both for the norm topology (Kuiper's theorem) and for ...
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52 views

Normalized tight frame that is not orthonormal

Does anybody know an example of a normalized tight frame (wavelet frame) that is not an orthonormal frame of $L^2( \mathbb{R})$? So in other words $\{\psi_{j,k}(x):=2^{j/2}\,\psi(2^j\,x-k)\}_{j,k \in ...
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291 views

Orthogonal complements of intersections of closed subspaces

Let $H$ be a Hilbert space and $H_1, \cdots, H_n$ be closed subspaces of $H$. $\mathbf{Question}:$ Is it always true that the orthogonal complement $(H_1\cap\cdots\cap H_n)^\bot$ of the intersection ...
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2answers
423 views

Is the residual spectrum of every power bounded operator contained in the open unit disk?

$\newcommand{\cH}{\mathcal{H}} \newcommand{\CC}{\mathbb{C}}$ Let $\cH$ be a Hilbert space. A linear operator $T: \cH \to \cH$ is said to be power bounded if $\sup_{n \geq 0} \|T^n\| < \infty$. If ...
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1answer
97 views

Is $\textbf{FHILB}$ locally regular?

Is the category, $\textbf{FHILB}$, of finite dimensional Hilbert spaces and linear maps locally regular, where `locally regular' is defined like this ...
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1answer
111 views

Inner product spaces without symmetry/hermitian axiom

Consider a vector space $X$ over $\mathbb R$ and a bilinear form $ \langle \cdot, \cdot \rangle : X \times X \rightarrow \mathbb R$. We assume furthermore that for any $x \in X$ there exists $y \in ...
3
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1answer
174 views

When does an irreducible unitary real representation remain irreducible after complexifying it?

Consider a unitary real representation of a Lie group $G$ over a real Hilbert space $\mathcal{H}_\mathbb{R}$ \begin{equation} \rho:G\rightarrow U(\mathcal{H}_{\mathbb{R}}) \end{equation} Taking the ...
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1answer
155 views

How many subspaces are generated by three or more subspaces in a Hilbert space?

In the book of G. D. Birkhoff "lattice theory", it is mentioned that there are 28 subspaces that can be obtained from three subspaces in general position in a Hilbert space (using intersections and ...
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1answer
366 views

Operator theory of the Hessian

How can I learn more about the operator theory of the Hessian? The Hessian of a function $u : \Omega \rightarrow \mathbb R$ over a domain $\Omega \subseteq \mathbb R^n$ is the matrix of second ...
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1answer
112 views

Orthogonal functions with shrinking support

This question is more or less a cross post of http://math.stackexchange.com/questions/1218660/orthogonal-functions-with-shrinking-support. Let $X$ be a metric space (compact, if it helps) and let $Y$ ...
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103 views

Boundedness of a Hilbert space projection map

Reading this recent thread I was reminded of a related problem I still haven't solved so I post it here in hopes of a positive result. Let $V_0 \subset H_0$ and $V_1 \subset H_1$ be separable ...
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1answer
315 views

Existence of a projection operator onto subspace of Hilbert space

Let $V \subset H$ be Hilbert spaces with a continuous, compact and dense imbedding. Let $\{w_j\}_j \subset V$ be a basis of $V$ and of $H$ (so finite linear combinitions are dense) which is not ...
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0answers
193 views

Weak topology on subsets of a normed space

I have few questions about the subsets of a normed space $X$ endowed with the weak topology. Let $E$ be such subset. When is the norm a continuous function on $E$? When is the metric induced by the ...
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1answer
102 views

Orthogonal compact operators on an infinite dimensional Hilbert space [closed]

How do I show that when $H$ is an infinite-dimensional Hilbert space we can find two compact positive operators $u,v$ with infinite dimensional image and $u \perp v$? This statement can be found at ...
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Decidabilty of the Hilbert lattice and quantum logic

What is known about the decidability of (first-order formulas in) the structure $(\mathcal{L}(H),\leq)$, where $\mathcal{L}(H)$ is the collection of all closed linear subspaces of a (separable) ...
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1answer
80 views

Injective inclusion map from RKHS function space to $L_p(\mu)$

Let $X$ be a measurable space, $\mu$ be a $\sigma$-finite measure on $X$, and $H$ be a separable reproducing kernel Hilbert space over $X$ with a measurable kernel $k$. At a certain part in a proof I ...
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1answer
177 views

The Maximal $\ell_2$ norm of a signed sum of vectors

Suppose we have $n$ vectors in $\mathbb{R}^n.$ Consider the signed sum of these vectors: $$U(s_1,\ldots,s_n)=s_1 v_1+s_2 v_2 + \ldots + s_n v_n$$ where $s_j$'s can only take values of $+1$ or $-1.$ I ...
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1answer
140 views

Non-closability of an operator

Let $a$ be a positive continuous function nowhere differentiable on $[0,1]$. The operator $T$ in $H:=L^2(0,1)\oplus L^2(0,1)$ defined by $$T(u_1,u_2) := (u_1' + au_2',0)$$ on $\textrm{Dom} \,T := ...
4
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1answer
387 views

Weak topology on a pre-Hilbert Space

Since there was essentially no answers on my previous question, I will ask a partial case of it, which is very easy to state. Let $\left(X,\left<\cdot,\cdot\right>\right)$ be an inner product ...
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Recontruction of the weak topolgy from the scalar product on a subset of a Hilbert Space

Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map ...
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69 views

Hilbert scales of covariance operators

Assume we have 2 covariance operators(positive definite trace class) $S$ and $T$ on Hilbert space $\mathcal H$ with corresponding eigenpairs $\{e_j,\lambda_j\}$ and $\{f_j,\lambda_j\}$. Assume that ...