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        Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

        8
        votes
        2answers
        345 views

        Harmonic oscillator in spherical coordinates

        It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry. More precisely, the operator $$-\frac{d^2}{dx^2}+x^2$$ can be ...
        5
        votes
        0answers
        87 views

        What are the 'wonderful consequences' following from the existence of a minimal dense subspace?

        In Peddechio & Tholens Categorical Foundations they quote PT Johnstone in their chapter on Frames & Locales: ...the single most important fact which distinguishes locales from spaces: the ...
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        vote
        0answers
        44 views

        Best constant for Hölder inequality in Lorentz spaces

        It's well known (and proved by R. O'neil) that there is a version of H?lder's inequality for Lorentz spaces, namely $$\|fg\|_{L^{p, q}} \lesssim_{p_1, p_2, q_1, q_2} \|f\|_{L^{p_1, q_1}}\|g\|_{L^{...
        2
        votes
        0answers
        79 views

        Anzai flow in noncommutative geometry

        Consider Anzai flows (cf. Anzai: Ergodic Skew Product Transformations on the Torus, Osaka Math. J. 3 (1951), 83-99) on the two dimensional torus $T^2$. I would like to know if there exists some ...
        2
        votes
        0answers
        71 views

        Dual of the space of affine functions

        Let $M^+(D)$ be the space of all positive measures on a closed convex subset $D$ of a locally convex topological vector space $E$. Two measure $\mu, \nu\in M^+(D)$ one can define a partial ordering $\...
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        votes
        0answers
        45 views

        Theta Summable operator with bounded trace

        Let $D$ be an unboudned self-adjoint operator on the Hilbert space $H$. We assume that all spectrum of $D$ are eigenvalues and $D$ is theta-summable, i.e. $e^{-tD^2}$ is of trace class for all $t>...
        4
        votes
        0answers
        110 views

        Cartan MASA and group measure space construction

        Let $N$ be a ${\rm II}_1$ factor. A maximal abelian self-adjoint subalgebra (MASA) is a $*$-subalgebra $A \subset N$ such that $A' \cap N = A$. It is called Cartan if $\mathcal{N}(A)''=N$, with the ...
        4
        votes
        2answers
        129 views

        Compact images of nowhere dense closed convex sets in a Hilbert space

        Let $B=-B$ be a nowhere dense bounded closed convex set in the Hilbert space $\ell_2$ such that the linear hull of $B$ is dense in $\ell_2$. Question. Is there a non-compact linear bounded operator ...
        2
        votes
        1answer
        54 views

        How does a statistical divergence change under a Lipschitz push-forward map?

        Let $\mu, \nu$ be two probability measures on a space $X$ (assume Polish space). $T: X \rightarrow Y$ is a Lipschitz-map that acts as a push-forward on these measures; let $\mu^\prime = T_{\#\mu}$ and ...
        0
        votes
        0answers
        21 views

        Specific Bounds on Divergence operator in Sobolev setting

        Given that $\vec{u}=(u_1,u_2)\in H^1(\Omega)\times H^1(\Omega)$ where $\Omega\subseteq\mathbb{R}^n$ is open and bounded with Lipschitz boundary, does there exist specific inequalities which links $\...
        4
        votes
        1answer
        130 views

        What is the trace of the integral operator $(\mathcal{L}f)(x)=\int_0^\infty (x \wedge y)f(y) \, d \pi(y)$?

        Let $\pi$ denote a probability measure on $[0,+\infty)$ and let us assume that $$m:=\int_0^\infty x \, \mathrm{d} \, \pi(x)<+\infty.$$ Let us consider the integral operator $\mathcal{L}$ on $L_2(\...
        6
        votes
        1answer
        114 views

        Density-$c_0$ in $\ell^\infty$

        Let $A \subseteq \mathbb{N}$, define the upper density of $A$ as, $$ \overline{\delta}(A) := \limsup_{N\to\infty}\frac{|A\cap\{1,2,3,\cdots,N\}|}{N}. $$ This naturally leads to a weaker form of ...
        1
        vote
        0answers
        111 views

        The perturbation of a convex function can also be convex?

        $ W^{1,\infty}(D)\ni f:D\to\mathbb R, (x,y)\mapsto f(x,y)$, is a strictly increasing on both dimensions (i.e. if $x_1>x_2$ then $f(x_1,y)>f(x_2,y)$), lipschitz continuous function defined on a ...
        0
        votes
        0answers
        27 views

        A fixed point in orderd cone

        While studing the fixed point in orderd fixed point I noticed that most of the time they worked in a Banach space orderd by a cone's order: Let $X$ be an ordered Banach space with order $\leq $. A ...
        22
        votes
        2answers
        916 views

        Intuition about L^p spaces

        I have read somewhere the following very nice intuition about $L^p(\mathbb{R})$ spaces. This graphic shows a lot of nice relations: 1) There is no inclusion between $L^p$ and $L^q$ 2) $L^p$ is the ...

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