11

The answer is yes (there exists a GT space whose dual is not a GT space), given by the very first test example that one might consider. The form of Grothendieck's theorem that gives rise to the terminology "GT-space" is the fact that every bounded operator from $\ell_1$ to $\ell_2$ is 1-summing. So the first test for your question would be: is ...


9

(reading "sequence" as "net", as suggested in the comments) Well, $C_b(\mathbb{R}^+) \cong C(\beta\mathbb{R}^+)$, so any such $L$ will arise from a probability measure on the Stone-Cech remainder $\beta\mathbb{R}^+ \setminus \mathbb{R}^+$. You need some choice principle to know this set is nonempty, so no example can be very explicit. (I ...


6

$\newcommand{\IR}{\mathbb{R}}$It is sufficient to consider $\theta=e_n$, the $n$-th standard basis vector, because the Sobolev norms are all rotationally invariant. For a compactly supported, smooth functions $f$: $$\begin{align*} \| \partial^k Rf(\cdot,\theta)\|_{L^1(\IR)} &= \int_\IR \left| \partial^k \int_{\IR^{n-1}} f(t\theta+y) dy \right|dt \\&= ...


5

As is so often the case, the isomorphism is the only reasonable map you can write down in the general case: $N^\perp / M^\perp \to (M/N)^\ast, f+M^\perp \mapsto (m+N \mapsto f(m))$ All that's left to prove is that it actually works ;-)


4

I expand my comment where I claim that, on the space (call it $N(V)$) of all norms on $V$, the smallest topology making continuous the evaluations $\|\cdot\| \mapsto \|v\|$ (for $v \in V$) coincides with the topology defined by the distance $d$. This clearly implies that, under your hypothesis, the maps $t \mapsto \|\cdot\|_t$ and $t \mapsto \|\cdot\|'_t$ ...


3

The fractional derivative $|\partial_x|^\alpha$ is discussed in One-dimensional wave turbulence by Zakharov, Dias, and Pushkarev. (Zakharov introduced the notation.) As they explain below Eq. 2.1, it is indeed defined via the Fourier transform, such that the Fourier transform of $|\partial_x|^\alpha\psi(x)$ is $|k|^\alpha\psi(k)$. Their appendix A contains a ...


2

This doesn't even hold for bounded operators. I will give a counterexample involving $2 \times 2$ matrices. Take $A_{+}= Id_2$, the $2 \times 2$ identity matrix. $A_{-}=0$ and $B= \begin{pmatrix} 1 &0\\0 &0 \end{pmatrix}$. Let $x_1=\begin{pmatrix} 1\\ -1 \end{pmatrix}$. Let $x_2=\begin{pmatrix} -2\\ -1 \end{pmatrix}$. Then $|x_2^tBx_1|=2> 1=|x_2^...


2

You can find many relevant results in the book: Pisier, Gilles Factorization of linear operators and geometry of Banach spaces. CBMS Regional Conference Series in Mathematics, 60. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. See, for example, Theorem 10.6 and ...


1

The answer depends on what you mean by "something similar". :-) The expression for the classical heat equation has the following probabilistic interpretation. Let $X_t$ be the $d$-dimensional Brownian motion, started at a given point $x$, and let $\mathbb P^x$ denote the corresponding probability. As a consequence of Itô's lemma, if $u$ is a ...


1

Just like studying bioinformatics programming is a good way for a non-biologist to get a basic knowledge of the fundamentals of molecular biology, studying quantum computing is a good way to get a basic knowledge of the fundamentals of quantum mechanics. The book Mathematics of Quantum Computing: An Introduction by Wolfgang Sherer (Springer 2019) gives a ...


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