# Questions tagged [fa.functional-analysis]

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

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### Openness of the set of injective functions in $C(\mathbb{R})$?

Let $C(\mathbb{R})$ be equipped with the topology of compact convergence (or equivalently the compact-open topology). Then, is the subset $\left\{f\in C(\mathbb{R}): \text{$f$injective} \right\}$ ...
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### Algorithm/iterative procedure for constructing hypercyclic vectors?

Let $B$ be a separable Banach space and let $L:B\rightarrow B$ be a hypercyclic operator; here I use the definition of hypercyclicity given implicitly by Birkhoff's Transitivity Theorem: continuous ...
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### Image restoration quality general lower bounds

A typical image restoration model posits that, starting from a true image $f = f(x,y)$, we observe $$\tilde f = f \star h + n$$ where $\star$ is convolution, $h$ is the point spread function (caused,...
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### Supremum over which sets makes $H^{\infty}$ non-separable?

It is known that the space $H^{\infty}$ of bounded holomorphic functions on the unit disk $D$ is non-separable with respect to the supremum norm $\|\cdot\|_{\infty}^{D}$. Let $E\subset D$ be connected ...
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### Error rate implying regularity

My question is a bit general/vague. It is well known that the regularity of certain functions can be measured through the rate of decay of certain error quantity based on an approximation procedure (...
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### What is known about the “unitary group” of a rigged Hilbert space?

Suppose that $(E,H)$ is a rigged (infinite dimensional, separable) Hilbert space, i.e. $H$ is a Hilbert space, and $E$ is a Fréchet space, equipped with a continuous linear injection $E \rightarrow H$ ...
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### Examples of amenable Banach algebras which have non-amenable subalgebra

I am looking for examples of amenable Banach algebras which have non-amenable subalgebra I know 1: Each amenable Banach algebra has a bounded approximate identity 2: If $I$ be a closed ideal in an ...
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### Using Paley-Wiener Theorem to prove the decay of $G(x-y)$

This question is related to my previous one, where I was looking for some help to prove the decay of the lattice Green function: \begin{eqnarray} G(x-y) = \int_{[-\pi,\pi]^{d}}\frac{d^{d}k}{(2\pi)^{d}}...
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### About the metrizability of the space of Probability measures $\mathcal{P}(S)$

It is often proved in Books that the space of Probability measures $\mathcal{P}(S)$ on a Polish metric space $(S,\rho)$ endowed with the weak/narrow topology induced by declaring it to be be the ...
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### Infinite composition of continuous functions

Let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ be a sequence of functions and define $F_n:= f_n\circ \dots\circ f_1$. Then $F_n$ is continuous. However, the pointwise limit need not be (consider Mateusz'...
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### On hereditarily reflexive Banach spaces

It was proved by W.B. Johnson and H.P. Rosenthal [Studia Math. 43 (1972), 77–92] that every Banach space $X$ with $X^{**}$ separable is hereditarily reflexive: every infinite dimensional closed ...
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### Logical axioms used in the construction of counterexamples to ISP

In many cases, some problems are either solved in an affirmative way, or in a negative way. However, in some cases it turns out that some logical axioms lead to a proof of a certain statement, while ...
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### A weaker weak time derivative than the one arising from Gelfand triples?

Let $V \subset H \subset V^*$ be a Gelfand/evolution triple where $V$ is a reflexive, separable Banach space and $H$ is a Hilbert space which has been identified with its dual. In the context of ...
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### Is $C(\mathbb{R}^n)$ is a DF-Space?

I recently have begun reading about DF-spaces and its clear to me that $C(K)$ is a DF-space for any compact subset (non-empty) $K$ of some $\mathbb{R}^D$ for finite D, since $C(K)$ is Banach. However,...
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### A norm inequality for operators

Let $A,B,C$ be self-adjoint operators of $L^2(\mathbb{R}^n)$ ($A$ and $B$ unbounded), $A\geq 0$, $B \geq 0$, with $\sqrt{A} C$ and $\sqrt{B} C$ bounded. Is the following inequality true for some ...
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### $\|(A_n-z)^{-1} - (A-z)^{-1}\|\to 0\;\Rightarrow\; \|e^{-tA_n}-e^{-tA}\|\to 0$ for general $C_0$ semigroups?

In short, the question is whether norm-resolvent convergence implies operator-norm convergence of the assocoated semigroups. More specifically, assume the following: The $A_n$ generate contraction ...
Let $H$ be a complex infinite dimensional separable Hilbert space. There are various extensions of the following well known result: Theorem (Lomonosov): Every nonscalar $T \in B(H)$ which commutes ...
Let $A$ be a von Neumann algebra. Then a classic observation is that the set of projections $\Pi(A)$ is naturally a complete orthomodular lattice. Question 1: Is the construction $A \mapsto \Pi(A)$ a ...