Homomorphisms on Function Algebras

1994 ◽  
Vol 46 (4) ◽  
pp. 734-745 ◽  
Author(s):  
M. I. Garrido ◽  
J. Gómez Gil ◽  
J. A. Jaramillo
Keyword(s):  
Banach Space ◽  
Topological Space ◽  
Wide Class ◽  
Rational Functions ◽  
Positive Answer ◽  
Algebra Homomorphism ◽  
Function Algebras ◽  
Real Analytic Functions ◽  
Real Analytic ◽  
Separable Spaces

AbstractLet A be an algebra of continuous real functions on a topological space X. We study when every nonzero algebra homomorphism φ: A → R is given by evaluation at some point of X. In the case that A is the algebra of rational functions (or real-analytic functions, or Cm-functions) on a Banach space, we provide a positive answer for a wide class of spaces, including separable spaces and super-reflexive spaces (with nonmeasurable cardinal).

scholarly journals Плюригармонические определимые функции в некоторых $o$-минимальных расширениях вещественного поля

2021 ◽  
pp. 35-40
Author(s):  
M. Berraho
Keyword(s):  
Harmonic Functions ◽  
Positive Answer ◽  
Real Field ◽  
Real Numbers ◽  
Pluriharmonic Function ◽  
The Real ◽  
Division Theorem ◽  
Minimal Structure ◽  
Real Analytic Functions ◽  
Real Analytic

In this paper, we first try to solve the following problem: If a pluriharmonic function $f$ is definable in an arbitrary o-minimal expansion of the structure of the real field $\overline{\mathbb{R}}:=(\mathbb{R},+,-,.,0,1,<)$, does this function be locally the real part of a holomorphic function which is definable in the same expansion? In Proposition 2.1 below, we prove that this problem has a positive answer if the Weierstrass division theorem holds true for the system of the rings of real analytic definable germs at the origin of $\mathbb{R}^n$. We obtain the same answer for an o-minimal expansion of the real field which is pfaffian closed (Proposition 2.6) for the harmonic functions. In the last section, we are going to show that the Weierstrass division theorem does not hold true for the rings of germs of real analytic functions at $0\in\mathbb{R}^n$ which are definable in the o-minimal structure $(\overline{\mathbb{R}}, x^{\alpha_1},\ldots,x^{\alpha_p})$, here $\alpha_1,\ldots,\alpha_p$ are irrational real numbers.

scholarly journals Characterizations and properties of graphs of Baire functions

2018 ◽  
Vol 68 (4) ◽  
pp. 789-802
Author(s):  
Balázs Maga
Keyword(s):  
Banach Space ◽  
Topological Space ◽  
Convex Set ◽  
Vertical Section ◽  
Fréchet Space ◽  
Frechet Space ◽  
Similar Question ◽  
Baire Functions

Abstract Let X be a paracompact topological space and Y be a Banach space. In this paper, we will characterize the Baire-1 functions f : X → Y by their graph: namely, we will show that f is a Baire-1 function if and only if its graph gr(f) is the intersection of a sequence $\begin{array}{} \displaystyle (G_n)_{n=1}^{\infty} \end{array}$ of open sets in X × Y such that for all x ∈ X and n ∈ ℕ the vertical section of Gn is a convex set, whose diameter tends to 0 as n → ∞. Afterwards, we will discuss a similar question concerning functions of higher Baire classes and formulate some generalized results in slightly different settings: for example we require the domain to be a metrized Suslin space, while the codomain is a separable Fréchet space. Finally, we will characterize the accumulation set of graphs of Baire-2 functions between certain spaces.

scholarly journals The Lagrangian Stability for a Class of Second-Order Quasi-Periodic Reversible Systems

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Yanling Shi ◽  
Jia Li
Keyword(s):  
Differential Equation ◽  
Analytic Functions ◽  
Order Differential Equation ◽  
Reversible Systems ◽  
Invariant Curves ◽  
Real Analytic Functions ◽  
All Solutions ◽  
Small Twist Theorem ◽  
Real Analytic ◽  
Twist Theorem

We study the following two-order differential equation,(Φp(x'))'+f(x,t)Φp(x')+g(x,t)=0,whereΦp(s)=|s|(p-2)s,p>0.f(x,t)andg(x,t)are real analytic functions inxandt,2aπp-periodic inx, and quasi-periodic intwith frequencies(ω1,…,ωm). Under some odd-even property off(x,t)andg(x,t), we obtain the existence of invariant curves for the above equations by a variant of small twist theorem. Then all solutions for the above equations are bounded in the sense ofsupt∈R|x′(t)|<+∞.

scholarly journals On Dentability in Locally Convex Vector Spaces

2017 ◽  
Vol 10 ◽  
pp. 14-19
Author(s):  
Oleg Reinov ◽  
Asfand Fahad
Keyword(s):  
Vector Space ◽  
Wide Class ◽  
Positive Answer ◽  
Vector Spaces ◽  
Bounded Subset ◽  
Locally Convex

The notions of V-dentability, V-s-dentability and V-f-dentability are introduced. It is shown, in particular, that if B is a bounded sequentially complete convex metrizable subset of a locally convex vector space E and V is a neighborhood of zero in E, then the following are equivalent: 1). B is subset V-dentable; 2). B is subset V-s-dentable; 3). B is subset V-f-dentable. It follows from this that for a wide class of locally convex vector spaces E, which strictly contains the class of (BM) spaces (introduced by Elias Saab in 1978), the following is true: every closed bounded subset of E is dentable if and only if every closed bounded subset of E is f-dentable. Also, we get a positive answer to the Saab's question (1978) of whether the subset dentability and the subset s-dentability are the same forthe bounded complete convex metrizable subsets of any l.c.v. space.

On zero sets of harmonic and real analytic functions

2018 ◽  
Vol 42 (2) ◽  
pp. 159-167
Author(s):  
André Boivin ◽  
Paul M. Gauthier ◽  
Myrto Manolaki
Keyword(s):  
Analytic Functions ◽  
Zero Sets ◽  
Real Analytic Functions ◽  
Real Analytic
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