# Questions tagged [fa.functional-analysis]

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

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### If $f \circ u \in BV$ and $f$ is strictly monotone, then is $u \in BV$?

Let $f: \mathbb R \to \mathbb R$ be a Lipschitz strictly monotone (so, in particular, invertible) function. Let $u: \mathbb R \to \mathbb R$. If $f \circ u \in BV$ can we conclude that $u \in BV$?
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### Orthogonal decomposition of $L^2(SM)$

I have been stuck on the following problem for a long time but I could not get the answer. Would you please help me? I was reading one paper [Paternian Salo Uhlmann: Tensor tomography on the surface] ...
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### Can Birkhoff's ergodic theorem for integrable functions easily be deduced from Birkhoff's ergodic theorem for bounded functions?

It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it ...
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### Confusion on the paper "Cohomology of maximal ideal space"

In the paper Cohomology of Maximal Ideal Space, there is a corollary about if $M$ is a compact orientable n-dimensional manifold, then $C(M,\mathbb{C})$ cannot be generated by fewer than n+1 elements. ...
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### Reference for Choquet-like theorem

While reading a paper, I encountered the following statement: Let $K$ be a convex compact subset of a locally convex topological vector space. If $\mu \in P(K)$ is a Radon probability measure on $K$, ...
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### Converse of mean value theorem almost everywhere? (Version 2)

Note: This is an attempt to narrow down conditions under which the conjecture stated in this previous post is true. As stated, it is false as shown by the counterexample provided in the answers by the ...
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### Nested nets of closed bounded star-shaped sets in a semi-reflexive space

Among Hausdorff locally convex spaces, semi-reflexivity is characterized by the weak topology having the Heine-Borel property. It follows that, in a semi-reflexive space, every nested net of closed ...
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### Is Hausdorffness a categorical property in the category of locally convex spaces?

I want to characterize Hausdorffness of a locally convex space only using categorical terms of the additive category LCS of locally convex spaces and continuous linear maps, i.e., terms like mono- or ...
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### Semi-linear elliptic problem, energy functionals, Fréchet derivatives and the Newton method in Banach spaces

Suppose $\Omega\subset\mathbb{R}^n$ is a regular open set, $f\in L^2(\Omega)$ and consider the following elliptic problem. $$-\Delta u + u=f'(u) , \;\;u_{|\partial \Omega}=0,$$ where $f'$ is the ...
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### Is the lattice of bounded Henstock Kurzweil integrable functions countably complete?

The set of HK integrable functions with an integrable upper bound $f$ forms a lattice, and satisfies the MCT and DCT. Does this mean that the lattice is countably complete? Indexing any countable set, ...
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Let $D_1$ be a domain with smooth boundary and assume that $D_1$ is a proper subset of $D_2$ which is itself a bounded domain in $\mathbb R^n$ with a smooth boundary. Assume also that $D_2\setminus ... 0answers 88 views ### Books on limiting properties of matrices with growing size This question has been posted on Math-Se previously. I am studying asymptotic properties of the Projection Matrix $$H_n=X'(X'X)^{-1}X$$ By the Gerschgorin disc theorem, the bounds on the ... 0answers 84 views ### Equivalence between notions of dynamical coupling as defined by Villani in his book Optimal Transportation: Old and New$\DeclareMathOperator\law{law}$In Villani's book he presents the following notions of dynamical couplings: Let$(X,d)$be a Polish space. A dynamical transference plan$\Pi$is a probability measure ... 0answers 50 views ### Characterization of inverse limits of finite-dimensional convex cones Consider a countable inverse system$C_1\substack{f_1 \\ \leftarrow} C_2 \substack{f_2 \\ \leftarrow} C_3 \substack{f_3 \\ \leftarrow} \ldots$where the$C_i$are finite-dimensional convex cones of ... 1answer 156 views ### Set where the speed of convergence is uniform in Lebesgue's density theorem Let$B \subset \mathbb R^n$be the unit ball. Consider a Borel measurable set$E \subset B$with positive Lebesgue measure$|E|>0$(say$|E| = |B|/2$). Then, Lebesgue's density theorem, says that ... 3answers 679 views ### Criterion for compactness Let$T:H\to H$be a continuous operator on a Hilbert space. Assume there exists an orthonormal base$(e_j)_{j\in\mathbb N}$, such that the sequence$Te_j$tends to zero. Must$T$be compact? 0answers 61 views ### When does an RKHS contain another? Consider a psd kernel function on the unit-sphere in$\mathbb R^d$off the form$K(x,x') = \varphi(x^\top x')$for some$\varphi:[-1,1] \to \mathbb R$, and let$\mathcal H_\varphi$be the induced ... 1answer 108 views ### Proving an estimate for the Neumann problem on$\mathbb{R}^3 \setminus B_1$in Weighted Sobolev spaces Let$M := \mathbb{R}^3 \setminus B_1$where$B_1$is the unit ball. It is known that for every$g \in H^{\frac{1}{2}} (\partial M)$, and for an appropriate$\delta$, there exists a unique solution$u$... 0answers 68 views ### Reference request: Optimal controls can be assumed to take values in a convex set Consider the deterministic controlled system: $$\dot x(t) = Ax(t) + Bu(t), \ t \in [0, T]$$ $$x(0) = x_0$$ where$x: [0, T] \to \mathbb R^n$is the controlled state process,$A \in \mathbb R^{n \times ...
Let $\{T_k\}_k, T$ be unbounded selfadjoint operators on a Hilbert space $H$. If $T_k\to T$ in the norm-resolvent sense, then for any $(a,b)\subset \mathbb R$ with $\{a,b\}\cap \sigma(T)=\emptyset$, ...