# Questions tagged [fa.functional-analysis]

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

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### The symmetric group of positive integers acting on $\ell^\infty(\mathbb{N})^*$

Let $\mathfrak{S}_\mathbb{N}$ be the symmetric group of all positive integers. Let $\ell^\infty(\mathbb{N})^*$ be the dual space of $\ell^\infty(\mathbb{N})$ equipped with weak*-topology. There is a ...
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### (Sharp) Bounds on $E(XYZ)$ given all the bivariate marginals

Suppose $X,Y,Z$ are all real-valued random variables. Suppose I know the joint marginal distributions of $(X,Y)$, $(Y,Z)$ and $(X,Z)$. I want to find bounds on $E(XYZ)$. In the case of bounding $E(XY)$...
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### Elliptic equation on square with periodic boundary values for the solution and it's partial derivatives

Suppose that $\Omega=[0,1]^2$. I will say that a real valued function $u$ on $\Omega$ satisfies periodic boundary values if $$u(x,0)=u(x,1), \;u(0,y)=u(1,y),\;\;\;\text{ for all }x,y\in[0,1].$$ Now ...
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### Elliptic regularity when the Lagrangian is possibly infinite

I want to solve variational problems of the form $$\inf_u \int_{-1}^1 \phi(u'(x)) \text{ with } u(-1)=u(1) = 0,$$ where $\phi(p)$ is convex and is allowed to take on the value $+\infty$ for some ...
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### A perturbation of an unconditionally convergent series in $\ell_2$

For two functions $x,y:\omega\to\mathbb R$ let $xy:\omega\to\mathbb R$, $xy:n\mapsto x(n)y(n)$, be their pointwise product. It is clear that for any elements $x,y\in\ell_2$ their pointwise product $xy$...
Let $X$ and $Y$ be Hilbert space, $A:X \to Y$ compact and injective, $(\sigma_n;v_n,u_n)$ be its singular value decomposition, that is, $$Av_n = \sigma_n u_n \\ A^* u_n = \sigma_n v_n$$ Since $\... 0answers 25 views ### Is these some transformation that transforms an arbitrary$\mathcal C^1$function into an$L$-smooth function? Let$\mathcal X\subset \mathbb R^d$. We say a function$f:\mathcal X\to \mathbb R$is$L$-smooth if it is continuously differentiable and$\|\nabla f(x)-\nabla f(y)\|\le \|x-y\|$holds for all$x,y\in ...
For infinite-dimensional Gaussian measures, we often see the definition of Gaussian random variables like this: Let $H(\Omega;\mathbb{R})$ be a separable Hilbert space. A random variable $u \in H$ is ...