# Questions tagged [fa.functional-analysis]

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

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### Examples/applications of parabolic PDEs that are not posed on domains or manifolds

Are there any examples of parabolic PDEs $$u' - Au = f$$ posed in a Gelfand triple setting $V \subset H \subset V^*$ with $V$ and $H$ chosen NOT as spaces of functions (or distributions) over a domain ...
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### Poisson summation formula for infinite dimensional spaces

Let $M$ be an orientable, compact smooth manifold with and $H^{-1}(M)$ be the dual space of $$H^{1}(M)=\{f:\int |f|^2+(\nabla f)^2<\infty\}$$ I know it is well known that (see Julien Dubedat, page ...
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### Kolmogorov entropy of a subset of $L^1$

How can we estimate the Kolmogorov $\epsilon$-entropy $$H_\epsilon (A,L^1(\mathbb R))$$ where $A = \{f:\mathbb R \to [0,K] \text{ s.t.$f \in L^1$and has total variation$TV(f) \le M$}\}$?
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### A strong duality for convex functional optimization that admits Lipschitz continuity constraints?

Problem Statement I am looking for formal proof---hopefully textbook material---of two items: an analogue to Slater's condition  that obtains strong duality for optimization of convex functionals; ...
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### A function $V : \mathbb{R}^2 \to \mathbb{R}$ is a (logarithmic) potential

I'm looking for references given some sort of inverse problem in logarithmic potential theory. That is, given a function $V : \mathbb{R}^2 \to \mathbb{R}$, what is a sufficient (and perhaps necessary) ...
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### holomorphy in infinite dimensions (holomorphic families of operators)

Let $X$ be a Banach space (over $\mathbb C$), and let $\mathcal L(X)$ be its algebra of bounded linear operators. Let $U\subset \mathbb C^N$ be an open subset, and $f:U\to \mathcal L(X)$ a function ...
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### Weak upper semi-continuity with orthogonal condition

Let $B \subset \mathbb{R}^d$ be the ball of radius one, and consider the map defined on $L^2(B,\mathbb{R})$ \begin{align*} f(\phi) = \underset{\substack{ \varphi \in H^1(B,\mathbb{R}) \\ \left<...
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### Scottish Book Problem 172

The problem is formulated using old terminology and I want to understand what it actually says. The problem reads: "A space $E$ of type (B) has the property (a) if the weak closure of an ...
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### Strong maximum principle for a PDE with coefficient in $L^1$

Let $U$ be a bounded open subset of $\mathbb{R}^n$ with smooth boundary, $n \geq 3$. Set $N = \frac{2n}{n-2}$. I am interested in the following equation: $$-\Delta \phi + R \phi + \phi^{N-1} = 0$$ ...
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### If $F$ is a countably normed, nuclear Fréchet space, can I then find a fundamental system which exhibits both of these properties at once?

Let $F$ a Fréchet space. This means that $F$ is a complete Hausdorff topological space whose topology can be generated by an increasing family of seminorms $\{ p_{n} \}_{n \in \mathbb{N}}$. Let's ...
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### Well-posedness of hyperbolic system with constant coefficients in finite domains

I'm studying the PDE $$\frac{\partial u}{\partial t} + A_x\frac{\partial u}{\partial x} + A_y\frac{\partial u}{\partial y} + A_z\frac{\partial u}{\partial z} = 0$$ with $A_x, A_y, A_z$ being ...
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### Inverting convolutions over finite intervals

There are well-known techniques for inverting convolutions over the whole or half real line with Fourier and Laplace transformations, but on the face of it they can't be applied to an integral ...
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### Weak convergence in a product space

Given a function $f: Y\longrightarrow Y$ ($Y$ is a Banach space). Assume that $f$ satisfies: If $y_n \rightharpoonup y$, then $f(y_n)\rightharpoonup f(y) \text{ in } Y$; $f$ is weakly compact; ...
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### Commutation between integrating and taking the minimal eigenvalue

Let $S = (f_{ij})_{ij}$ be a $n \times n$ real symmetric matrix, with functions $f_{ij} \in L^1(\mathbb{R}^d,\mathbb{R})$ in it. We define $\left(\int u S \right)_{ij} = \int u S_{ij}$ as the ...
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### Hamiltonian dynamics on cotangent bundle

I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $N$ is a differential manifold and $S_0 ,S_1\subseteq N$ two ...