# Questions tagged [fa.functional-analysis]

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

5,346 questions
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### Weak* convergence in a dual Banach lattice vs norm convergence of moduli

Let $E$ be a dual Banach lattice, that is, $E = E_*^*$ for some Banach lattice $E_*$ (I have $M(X)=C(X)^*$ specifically in mind for a compact space $X$). Suppose that $(x_n)$ is a weak*-null ...
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### Weak*-convergence of signed measures

Let $X$ be a compact Hausdorff space and let $M(X)$ denote the space of signed measures that is naturally dual to $C(X)$, the space of continuous functions on $X$. I am interested whether the ...
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### Unboundedness of the Sobolev norm of a sequence : Does it follow from Sobolev embedding?

Let $\{f_n\}$ be a sequence of functions that are continuous and lying in $H^k(\mathbb{R}^m)$. Assuming $k>\frac{m}{2}$, and if $f_n \to f$ pointwise, where $f\in H^k(\mathbb{R}^m)$, such that $f$ ...
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Problem. Is there any complemented subspace in the Banach space $C(\beta\mathbb N\times\beta\mathbb N)$, not isomorphic to $c_0$, $c_0\oplus C(\beta\mathbb N)$, $C(\beta\mathbb N)$, $c_0(C(\beta\... 0answers 120 views ### A basis of the Banach space$L^p(\mathbb T^\omega)$consisting of characters Problem: For$1<p<\infty$,$p\ne 2$, has the complex Banach space$L^p(\mathbb T^\omega)$got a Schauder basis consisting of characters of the compact topological group$\mathbb T^\omega$? (... 0answers 50 views ### Characterizing the separability of the Gelfand space of a semisimple commutative Banach algebra Problem. Is the separability of the Gelfand space of a semi-simple commutative Banach algebra$A$equivalent to the existence of a countable family$\{\varphi_n\}_{n\in\omega}$of multiplicative ... 3answers 132 views ### Reconstructing a curve in$S^2$from intersections with great circles Take$S^2$with its standard metric. The space of great circles in$S^2$can be identified with the real projective plane$\mathbb{R}P^2$. Let$X$be an embedded circle in$S^2$; associate to it a ... 0answers 91 views ### An exponential integral over sphere I'm wondering if someone knows if the following kind of integral has a closed-form expression (or maybe an approximation): $$\int_{0}^1 x^{2a} (1-x^2)^b e^{-x^2 c} dx$$ for$a,b,c > 0$. [This has ... 0answers 61 views ### Is there an option to handle Neumann-series when it diverges? (using infinite-sized Carleman matrices) (I asked this in MSE but did not find resonance, there is also a relation to an older discussion here on summability see here and a followup formulating an$\text{ais}()$already here) ... 2answers 446 views ### Is taking the positive part of a measure a continuous operation? Here is question I tried to answer for some time - it seems to be straightforward, but I have trouble figuring it out. Let$\Omega$be a compact domain in$\mathbb{R}^n$. For any signed Borel measure ... 1answer 113 views ### A specific problem on : Can bounding the Sobolev norm, bound a higher derivative? Let$f \in H^k(\mathbb{R}^m)$,$k>\frac{m}{2}$. Given any$f$, such that$\|f\|_{H^k(\mathbb{R}^m)}<K$, and any$\phi \in C^{\infty}(\mathbb{R}^m)\cap H^k(\mathbb{R}^m)$, such that$\|\phi\|_{...
We have the abstract evolution equation $$U^{\prime}=A(t)U+F(U), \quad U(0) = U_0.$$ If the operator $A$ is independent of time, we can get local existence of solutions by proving that $A$ is the ...