# Questions tagged [fa.functional-analysis]

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

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### Order bounded version of monotone complete $C^*$-algebras

Let $A$ be a $C^*$-algebra with self-adjoint part $A_{\operatorname{sa}}$. Then $A$ is called monotone complete if every increasing norm bounded net in $A_{\operatorname{sa}}$ has a supremum (with ...
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### Interpolating sequences are strongly separated

I am reading Agler and McCarthy's Pick Interpolation and Hilbert Function spaces. In Chapter 9, titled "Interpolating Sequences", the authors say that every interpolating sequence is ...
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### A fractional weighted Poincaré inequality

Does there exists a constant $C>0$ such that $$\int_{-1}^1 \lvert x\rvert\lvert\partial_x u\rvert^2 \,dx \geq C\, \lVert u\rVert^2_{H^{1/2}((-1,1))},$$ for all $u\in C^{\infty}_0((-1,1))$?
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### Definition and properties of tangent functional

I am reading Measures Which Agree on Balls by Hoffmann-Jørgensen and I am somewhat confused. Here, $E$ is a Banach space, $S$ is the unit sphere, and $x \in S$. We let $\tau(x, \cdot)$ denote the ...
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Let $(M,g)$ be a compact Riemannian manifold and $\Delta_g$ its Laplace operator. Let $\varphi$ be a test function on $\mathbf{R}_{>0}$. We define the operator on, say, $L^2(M)$ $$T_{\varphi}(u) :=... 1 vote 1 answer 120 views ### Compare the weight of p\vee q and that of p+q Let M be a von Neumann algebra. If it has a semifinite faithful normal trace \tau, then we have \tau(p\vee q)\le \tau(p)+\tau(q). However, for the weight (even a faithful normal state) \omega ... • 617 1 vote 1 answer 73 views ### Bounded C_0-semigroups on barrelled spaces are equicontinuous I have the following question: Let X be a barrelled locally convex space (every absolutely convex, absorbing and closed set is a neighborhood of zero) and let (T(t))_{t\geq0} be a C_0-semigroup, ... • 129 1 vote 0 answers 81 views ### Dependence of Sobolev embedding theorem constant on smoothness Assume that \Omega \subset \mathbb{R}^d is "nice" enough and k is a positive real number. Using the Sobolev embedding theorem, we can get that$$ \|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...
This is a general question, more of a reference request. tl;dr: Is there a "calculus" for computing metric entropy bounds? Given a function space $\mathcal{F}$, we may define its metric ...