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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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        8 views

        What are the sets on which norm-closedness implies weakly closedness?

        Let $X$ be a Banach space. Let $C$ be a convex, and normed-closed subset of $X$. It is well-known that $C$ becomes weakly closed subset of $X$. I want to know is there any well-know class of non ...
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        1answer
        43 views

        Largest ideal in bounded linear maps on Schatten-$p$ class

        Let $1\leq p<\infty.$ Denote $S_p(\ell_2)$ be the set of all compact operator $x$ on $\ell_2$ such that $Tr(|x|^p)<\infty.$ Define $\|x\|_{S_p(\ell_2)}:=Tr(|x|^p)^{\frac{1}{p}}.$ This makes $S_p(...
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        25 views

        Lipschitz function of independent subgaussian random variables

        This question was asked here, but I have reason to believe that it's a serious research question appropriate for this forum (also, the answers given at the link aren't satisfactory). ?If $X\in\mathbb{...
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        108 views

        Reference Request: Stone-Weierstrass on Other Topologies

        Let $X$ be a compact subset of $\mathbb{R}^n$. The classical Stone-Weierstrass theorem describes dense subsets of $C(\mathbb{R}^n,\mathbb{R})$ when it is equipped with the compact-open topology. ...
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        1answer
        94 views

        Question about pointwise convergence of operators

        Consider two Banach spaces $E,F$ and a net $T_\alpha : E \to F$ of continuous operators. I know that for each $x \in E$ the net $T_\alpha (x)$ is convergent in $F$ and it is easy to show that the ...
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        60 views

        weaky compact subset of Banach space with separable predual

        Let $X$ be a Banach space and $S\subseteq X$ be a subspace such that the unit sphere of $S$ is weakly compact. If $Y^*=X$ for some separable Banach space $Y,$ is it true that $S$ is separable?
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        1answer
        219 views

        Ideal of strictly singular operators

        Let $X$ be a Banach space. An operator $T:X\to X$ is called strictly singular iff for any infinite dimensional subspace $Y\subseteq X,$ $T|_{Y}:Y\to T(Y)$ is not an isomorphism. It is known that for $...
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        42 views

        Spectral radius of a sequence of operators

        In this question $E$ stands for a complex Hilbert space and let $\mathcal{L}(E)$ the algebra of all bounded linear on $E$. Let ${\bf T}=(T_1,\cdots,T_d) \in \mathcal{L}(E)^d$ be an operator tuple. ...
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        1answer
        184 views

        Reference request: norm topology vs. probabilist's weak topology on measures

        Let $(X,d)$ be a metric space and $\mathcal{M}(X)$ be the space of regular (e.g. Radon) measures on $X$. There are two standard topologies on $\mathcal{M}(X)$: The (probabilist's) weak topology and ...
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        57 views

        Are interpolation polynomials uniformly bounded?

        How can prove a sequence of Lagrange polynomials ${{I_Nx}}$ that bounded at CGL points, is uniformly bounded?
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        1answer
        71 views

        The optimal asymptotic behavior of the coefficient in the Hardy-Littlewood maximal inequality

        It is well-known that for $f \in L^1(\mathbb{R^n})$,$\mu(x \in \mathbb{R^n} | Mf(x) > \lambda) \le \frac{C_n}{\lambda} \int_{\mathbb{R^n}} |f| \mathrm{d\mu}$, where $C_n$ is a constant only depends ...
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        0answers
        46 views

        Regularity for Laplacian operator on non-compact manifold

        Let $(M,g)$ be a complete non-compact Riemannian manifold . Thanks to @EveryLT, we know that the Poisson equation $$\Delta u=f,$$ is solvable for some $f\in L^2_k(M)$. Q Suppose that $(M,g)$ is ...
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        votes
        1answer
        121 views

        Luxemburg norm as argument of Young's function: $\Phi\left(\lVert f \rVert_{L^{\Phi}}\right)$

        Let $\Phi$ be a Youngs's function, i.e. $$ \Phi(t) = \int_0^t \varphi(s) \,\mathrm d s$$ for some $\varphi$ satifying $\varphi:[0,\infty)\to[0,\infty]$ is increasing $\varphi$ is lower semi ...
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        1answer
        62 views

        Invariance of Finite Dimensional Woronowicz $\mathrm{C}^*$-ideals under the Antipode

        Let $(A,\Delta)=:F(G)$ be a finite dimensional $\mathrm{C}^*$-Hopf algebra, and so the algebra of functions on a quantum group $G$. Let $J$ be a closed ideal in $F(G)$ and $\pi:F(G)\rightarrow F(G)/J$...
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        1answer
        115 views

        Spectral asymptotics of normal Hilbert-Schmidt operators

        Does anybody know a reference for the following theorem? Let $G \subset \mathbb{R}^m$ be open and of finite measure and $T \in L^2(G) \rightarrow L^2(G)$ be linear and bounded such that $$ R(T) \...

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