# Tagged Questions

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

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### Generalized Poincaré Inequality on H1 proof.

let's see if someone can help me with this proof. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. And let $L^2\left(\Omega\right)$ be the space of equivalence classes of square integrable ...
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### Norm of infinite dimensional Hilbert space to calculate difference between string lengths

I am trying to wrap my head around Proposition 13, last para, page 1049 in this paper. The authors are trying to prove certain properties of string edit distance (defined at the start of Section of 6....
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### A bounded linear functional on a Hilbert space that is a Hahn-Banach extension of one on a subspace

Let $M$ be a closed linear subspace of a Hilbert space $H$ and $g\in M*$(all bounded linear functional on $M$). Let $\pi$ be the orthogonal projection of H onto M, then $f=g\circ\pi$ is the only Hahn-...
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### Weakly square summable series as operators on Hilbert spaces

Let $H$ be a Hilbert space and let $\{a_n\}_{n\in\mathbb{N}}$ be a sequence in $H$ such that $\sum^{\infty}_{n=1}|\langle h,a_n\rangle|^2<\infty$ for all $h\in H$. Here $\langle\dot{},\dot{}\rangle$...
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### GNS-Construction: Involution

Given a C*-algebra $\mathcal{A}$. (It may or may not contain identity!) Consider a positive linear functional: $$\omega:\mathcal{A}\to\mathbb{C}:\quad A\geq0\implies \omega(A)\geq0$$ Construct its ...
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### References to: If $C\subset\mathbb{R}^n$ is convex and $0\notin C$ then there exists $v\in C$ such that $C$ is in the closed halfspace $H_v$.

For each $v\in\mathbb{R}^n$, we define the notation $H_v=\{u\in\mathbb{R}^n:\langle u,v\rangle\geq0\}$, where $\langle\cdot,\cdot\rangle$ denotes the usual inner product in $\mathbb{R}^n$. Recently, ...
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### Von Neumann algebraic Quantum Object is direct sum of type I factors

I am looking at the non-standard quantum projective spaces $A:=\mathcal{A}_q(\mathbb{CP}^n(c,d))$ introduced by Dijkhuizen and Noumi. Now I want to show that if I take the von Neumann algebra ...
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### Product of Lebesgue measure on Hilbert cube doesn't satisfy doubling condition?

The Hilbert cube $H$, is the infinite dimensional product $[0,1]\times [0,\frac12]\times...$ Let $\mu$ be product of Lebesgue measures $\mathcal{L}^1 \times \mathcal{L}^1\times...$, I heard that the ...
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### Functionals taking real values

Suppose $f$ is a bounded functional on a separable Hilbert space. Can we always find an orthonormal basis such that $f$ takes real values on that basis?
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### How to prove that $e_{\lambda}$ can be written in the following form?

Let $e_{\lambda}$ be the spectral density associated to the spectral function $E_{\lambda}$ for a self-adjoint operator $A$ on a complex Hilbert space $(H,\left<., .\right>)$. Haw to prove that ...
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### Question on Hermite functions

Stein - Real Analysis p.205 Hermite functions $h_k(x)$ are defined by the generating identity $\sum_{k=0}^\infty h_k(x)\frac{t^k}{k!} = e^{-x^2/2 + 2tx - t^2}$. I have proven that it satisfies ...
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### Property of Conical Hull

Let $H$ be a real Hilbert space and $C$ be a nonempty convex subset of $H$. The conical hull of $C$ is defined by $$\operatorname{cone}{C} := \bigcup_{\lambda >0}{\lambda C}.$$ (it is a cone in ...
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### What is “Bra” and “Ket” notation and how does it relate to Hilbert spaces?

This is my first semester of quantum mechanics and higher mathematics and I am completely lost. I have tried to find help at my university, browsed similar questions on this site, looked at my ...
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### Spectrum of Laplace operator with potential acting on $L^2(\mathbb R)$ is discrete

Consider an operator $H=-\Delta +U(x)$ on $L^2(\mathbb R)$ for a function $U(x): \mathbb R \to \mathbb R$ that tends to $+\infty$ as $|x|$ grows. These kinds of operators appear all over non-...
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In my studies of RKHS i.e. Reproducing Kernel Hilbert Spaces stating the following Let $\mathbb{H}$ be a RKHS on a set X. We are asked to characterize when the following set of kernels $\{ k_{... 1answer 18 views ### some detail calculation on the proof of equivalence of norms We say that two norm$\|x\|_1$and$\|x\|_2$on a vector space$X$are said to be equivalent if there exists$K>0$and$M>0$such that $$K\|x\|_1\le \|x\|_2\le M\|x\|_1$$ Prove that on a ... 3answers 76 views ### Why are function spaces generally infinite dimensional The other day, I was trying to explain some concepts in Fourier analysis and wavelets to my girlfriend (an electrical engineering student) and obviously, the concept of Lebesgue integration came up in ... 1answer 48 views ### Approximate point spectrum of a normal operator how can I show the following theorem? Let$H$a Hilbert space and$T:H \to H$a linear, continuos and normal operator. Then for every$\lambda \in \sigma(T)$there exists a sequence$(x_n)_{n \in \...
Let $K$ be a positive operator on a Hilbert space $H$. $Q_1$ and $Q_2$ are projections such that $Q_1\perp Q_2$. Is  E^{Q_1K Q_1} (1,\infty) + E^{Q_2K Q_2} (1,\infty) =E^{Q_1K Q_1 +Q_2K Q_2} (1,\...
### Two definitions of the operator $\exp(x)$ in $L^2(\mathbb R)$
The operator $x$ acts on a dense subspace of $L^2(\mathbb R)$ and is not bounded. So if we define $\exp(x)$ via the power series $\sum_{n=0}^\infty \frac {x^n}{n!}$, convergence will not follow in the ...