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        Questions tagged [fa.functional-analysis]

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

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        votes
        0answers
        39 views

        Functional characterization of local correlation matrices?

        Definition: A matrix $C\in\mathbb R^{m\times n}$ is local correlation matrix iff there exists real random variables $x_1,\dots,x_m,y_1,\dots,y_n$ defined on a common probability space which takes ...
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        votes
        2answers
        155 views

        Properties of heat equation

        ** I simplified the question: ** On bounded domains, the maximum principle implies that the solution to the heat equation is (strictly) positive, if the initial and boundary data is positive. I ...
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        votes
        0answers
        58 views

        Measurability of the heat semigroup in $L^\infty$

        Let $S(t)$ be the $C_0$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega)$, where $\Omega$ is a bounded open subset of $R^n$. It is known that $S(t)$ ...
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        votes
        0answers
        22 views

        Prove that the Bynum space $\ell ^{p, \infty}$ is a uniformly not square space

        In the book Measures of Noncompactness in Metric Fixed Point Theory, the authors prove that $\varepsilon_{0}(\ell ^{p, \infty}) = 1$, where $\ell ^{p, \infty} = (\ell ^{p}, \Vert \cdot \Vert _{p, \...
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        votes
        0answers
        121 views

        Largest weak(-like) topology with respect to which continuous functions are dense in the space of Borel functions

        Let $X$ denote the space of bounded Borel functions $f\colon [0,1] \to \mathbb{R}$. Let $M$ denote the space of finite Borel measures on $[0,1]$. What is the largest family $F \subset M$ such that for ...
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        votes
        1answer
        172 views

        Compactness of set of indicator functions

        Let $\chi_A(x)$ denote an indicator function on $A\subset [0,1]$. Consider the set $$K=\{\chi_A(x): \text{ A is Lebesgue measurable in }[0,1]\}.$$ Is this set compact in $L^\infty(0,1)$ with respect ...
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        vote
        0answers
        46 views

        About a class of expectations

        Consider being given a $n-$dimensional random vector with a distribution ${\cal D}$, vectors $a \in \mathbb{R}^k$, $\{ b_i \in \mathbb{R}^n \}_{i=1}^k$ and non-linear Lipschitz functions, $f_1,f_2 : \...
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        votes
        1answer
        167 views

        Can the characteristic function of a Borel set be approached by a sequence of continuous function through a certain convergence in $L^\infty$?

        Want to find $f_n$ a sequence of continuous functions, so that for all Borel regular measure $\mu$, we have $\int f_n d\mu\rightarrow\int \chi_\Delta d\mu$, as $n$ goes to infinity, where $\Delta$ ...
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        votes
        0answers
        29 views

        Is every pair of closed linear subspaces boundedly regular?

        In the context of the development of the spectral theory, J. v. Neumann showed that for every pair of closed linear subspaces $M,N$ of a Hilbert space the iterations $(P_M P_N)^nx$, where $P_M$ and $...
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        votes
        0answers
        28 views

        Function classes with high Rademacher complexity

        My question is two fold, Is there any general understanding of what makes a function class have high Rademacher complexity? (Sudakov minoration would say that one sufficient condition for a class of ...
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        votes
        1answer
        102 views

        Brascamp-Lieb inequalities on the sphere

        In the paper [CLL], Carlen, Lieb, and Loss demonstrate a version of the Young inequality on the sphere $S^{N-1}$ in $\mathbb{R}^N$. For positive functions $f_j$ on $[-1,1]$, the following bound holds:...
        2
        votes
        1answer
        59 views

        Existence of a bounded operator which satisfies the discrete product rule

        Is there a bounded self-adjoint operator $H$ acting on $\ell^2(\mathbb{Z})$ such that for all sequences $u,v\in \ell^2(\mathbb{Z})$ $$ H(uv)=(Hu)v+u(Hv)$$ where uv is the pointwise product. This is ...
        3
        votes
        1answer
        100 views

        Ultraproduct of non-commuative $L^p$-spaces

        Let $1<p<\infty.$ Let $I$ be a non-empty set and $\mathcal{U}$ be an ultrafilter over $I.$ Let $M_i$ be von Neumann algebras equipped with normal faithful semifinite traces $\tau_i,$ $i\in I.$ ...
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        votes
        0answers
        38 views

        Approximate a Lipschitz function by an affine function [closed]

        Suppose $\mathcal{F}$ is a Lipschitz function (Lipschitz constant $L$) space defined on $A$. $A$ is a bounded subset of $\mathbb{R^n}$. The $\mathcal{G}$ is an affine function space defined on $A$. ...
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        votes
        0answers
        51 views

        How to prove the binary function uniformly boundary?

        Assume that $a_1<1$, $a_3<1,$ $a_1+a_2+a_5>1$, $a_3+a_4+a_5>1,$ $a_1+a_2+a_3+a_4+a_5>2.$ For $x,y>0,$ define a fucntion $$H(u,v)=\frac{u^{\frac{1}{2}}\int_0^{\infty}\int_0^{\infty}\...

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