# Questions tagged [fa.functional-analysis]

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

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### Second dual $X^{**}$ of ternary $C^*$-ring $X$ is again ternary $C^*$-ring?

Recall that a ternary $C^*$-ring is a complex Banach space $X$, equipped with a associative ternary product $[.,.,.]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle ...
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### Hereditarily primary Banach spaces

A Banach space $X$ is said to be prime if every infinite dimensional complemented subspace is isomorphic to the space $X$. The space $X$ is primary if it has an infinite dimensional subspace $Y$ such ...
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### How to prove the second Korn inequality?

$\textbf{Theorem}.1$ (The first Korn inequality) Suppose that $\Omega$ is a bounded domain in $\mathbb{R}^d$ with Lipschitz boundary. Then\ \begin{eqnarray} \sqrt{2}\left\|\triangledown u\right\|_{...
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### limit of Riemann-Stieltjes sums as an integral on $\mathscr{H}$

I was reading Leon Takhtajan's book on quantum mechanics and, at some point, he states the J. von Neumann Theorem. The first part of this theorem is as follows. For every self-adjoint operator $A$ on ...
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### Rabinowitz global bifurcation

Do you have any (modern) references to recommend to learn the proof of the so called Rabinowitz Global Bifurcation Theorem? The original paper is not self-contained and I don’t find it very ...
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### Surjective linear isometries on $\ell_\infty(\mathbb{N})$

In Volume 1 of "Classical Banach Spaces" Lindenstrauss and Tzafriri note that all surjective linear isometries on $\ell_\infty$ are of the from $(a_i) \mapsto (\varepsilon_i a_{\pi(i)})$ ...
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### Haar integral of rational function of unitaries

I'm trying to compute the following Haar integral over the unitary group: $$\int\limits_{\mathbb{U}(d)}\dfrac{1}{\sum_{k,l=1}^d u_{ik}\overline{u_{il}}c_{kl}}dU.$$ Is there anything known about the ...
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### Is the derivative the unique operation on points in the plane that preserves convexity?

Let $C(n)$ be the space of multisets of size $n$ of points in the Euclidean plane, topologised appropriately, and consider a surjective continuous map: $$D:C(n)\rightarrow C(n-1)$$ Such that the ...
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### When is a product of two two-parameter Mittag-Leffler functions a Mittag-Leffler function?

I am studying properties of the two-parameter Mittag-Leffler function. $$E_{\alpha,\beta}(z)=\sum_{k=0}^\infty \dfrac{z^k}{\Gamma(\alpha k+\beta)}.$$ I am particularly interested in recurrences and ...
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Is there a definition Df(g) of uniform continuity of g, without using the notion of metric? Let $(E,d_E)$ and $(F, d_F)$ metrics spaces, $f$ continuous fonction of $E$ to $F$ We must have : Df$(f)$ ...
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### On the operator $f\to xf'/f$

I'm interested in the following operator $T$, close relative of the standard logarithmic derivative: $$f(x)\to Tf(x)=\frac{\text{d}(\log {f})}{\text{d}(\log {x})}=\frac{xf'}{f},$$ where $f$ is an ...
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### Algebra properties regarding Gevrey spaces: closed under multiplication

In page 24 of the paper Landau Damping: Paraproducts and Gevrey Regularity, the authors claimed an algebra property of Gevrey spaces, the formula (3.14), without giving a proof. So I'm asking for a ...
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### Must a Schauder basis for $W^{1,p}_0(\Omega)$ be oscillatory?

Suppose that $\Omega \subset \Bbb R^d$ is a sufficiently nice domain. From the examples of orthogonal bases in Hilbert space cases (or looking at a wavelets basis), it seems natural to me that one may ...
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### On weak Hahn-Banach smoothness

Let us recall Phelp's property-$U$: A subspace $Y\subset X$ is said to have property-$U$ if every $y^*\in Y^*$ has unique norm preserving extension over $X$. $Y$ is weak Hahn-Banach smooth if $y^*$ ...
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### A sequence of functions solving $-\Delta u_n + V u_n = u_{n-1}|_{\partial M}$

Let $M = \mathbb R^3 \setminus B_1$ where $B_1$ is the unit ball. Let $h \in C^{\infty}(\partial M)$ and let $u_0$ be the unique function that vanishes at infinity and solves \begin{cases} -\Delta ...
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### Different smooth structures on the infinite jet bundle (for the purposes of calculus of variations)

Let $\pi:Y\rightarrow X$ be a (smooth, finite dimensional) fibred manifold. Since no other fibrations will be considered on $Y$, I will identify $(Y,\pi,X)$ with $Y$. The finite order jet bundles are ...
For a Banach space $B$, its one-sided Unconditional Martingale Difference (UMD) constant $C^-_p$ (for $p \in (1,\infty)$) is the smallest value such that for all $B$-valued martingale difference ...