Maximal Ideals in Space of Continuous Functions
Intersections of Primary Ideals in Rings of Continuous Functions
1972 ◽
Vol 24 (3) ◽
pp. 502-519 ◽
Let C be the ring of all real valued continuous functions on a completely regular topological space. This paper is an investigation of the ideals of C that are intersections of prime or of primary ideals.C. W. Kohls has analyzed the prime ideals of C in [3 ; 4] and the primary ideals of C in [5]. He showed that these ideals are absolutely convex. (An ideal I of C is called absolutely convex if |f| ≦ |g| and g ∈ I imply that f ∈ I.) It follows that any intersection of prime or of primary ideals is absolutely convex. We consider here the problem of finding a necessary and sufficient condition for an absolutely convex ideal I of C to be an intersection of prime ideals and the problem of finding a necessary and sufficient condition for I to be an intersection of primary ideals.
2019 ◽
Vol 20 (1) ◽
pp. 109 ◽
<p>For any completely regular Hausdorff topological space X, an intermediate ring A(X) of continuous functions stands for any ring lying between C<sup>∗</sup>(X) and C(X). It is a rather recently established fact that if A(X) ≠ C(X), then there exist non maximal prime ideals in A(X).We offer an alternative proof of it on using the notion of z◦-ideals. It is realized that a P-space X is discrete if and only if C(X) is identical to the ring of real valued measurable functions defined on the σ-algebra β(X) of all Borel sets in X. Interrelation between z-ideals, z◦-ideal and Ʒ<sub>A</sub>-ideals in A(X) are examined. It is proved that within the family of almost P-spaces X, each Ʒ<sub>A</sub> -ideal in A(X) is a z◦-ideal if and only if each z-ideal in A(X) is a z◦-ideal if and only if A(X) = C(X).</p>
Transferring Results From Rings of Continuous Functions to Rings of Analytic Functions
1975 ◽
Vol 27 (1) ◽
pp. 75-87 ◽
Let C(X) be the ring of all real-valued continuous functions on a completely regular topological space X, and let A﹛Y) be the ring of all functions analytic on a connected non-compact Riemann surface F. The ideal theories of these two function rings have been extensively studied since the fundamental papers of E. Hewitt on C﹛X)[12] and of M. Henriksen on the ring of entire functions [10; 11]. Despite the obvious differences between these two rings, it has turned out that there are striking similarities between their ideal theories. For instance, non-maximal prime ideals of A (F) [2; 11] behave very much like prime ideals of C﹛X)[13; 14], and primary ideals of A(Y) which are not powers of maximal ideals [19] resemble primary ideals of C(X) [15]. In this paper we show that there are very good reasons for these similarities. It turns out that much of the ideal theory of A (Y) is a special case of the ideal theory of rings of continuous functions. We develop machinery that enables one almost automatically to derive results about the ideal theory of A(Y) from corresponding known results of ideal theory for rings of continuous functions.
Pointwise Sequentially Closed Ideals in C*(X)
1973 ◽
Vol 16 (1) ◽
pp. 115-117 ◽
The purpose of this paper is to determine the conditions under which the maximal ideals of the ring C*(X)—the bounded real-valued continuous functions on a completely regular Hausdorff space X—are closed under pointwise convergence of sequences. Whereas the maximal ideals of C*(X) are closed under pointwise convergence of nets if and only if X is compact, it is shown that a necessary and sufficient condition for their pointwise sequential closure is that X be pseudocompact (i.e. that all real-valued continuous functions of X be bounded).
Ideal convergence of nets of functions with values in uniform spaces
2017 ◽
Vol 31 (20) ◽
pp. 6281-6292
We consider the pointwise, uniform, quasi-uniform, and the almost uniform I-convergence for a net (fd)d?D of functions from a topological space X into a uniform space (Y,U), where I is an ideal on D. The purpose of the present paper is to provide ideal versions of some classical results and to extend these to nets of functions with values in uniform spaces. In particular, we define the notion of I-equicontinuous family of functions on which pointwise and uniform I-convergence coincide on compact sets. Generalizing the theorem of Arzel?, we give a necessary and sufficient condition for a net of continuous functions from a compact space into a uniform space to I-converge pointwise to a continuous function.
AbstractThe investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space $$\mathcal {C}(\Omega )$$ C ( Ω ) . Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions $$\mathcal {C}(\mathcal {T},E)$$ C ( T , E ) , where $$\mathcal {T}$$ T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions $$\mathcal {C}_0(\mathcal {T},E)$$ C 0 ( T , E ) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.
A necessary and sufficient condition in terms of lower cut sets are given for the insertion of a contra-continuous function between two comparable real-valued functions on such topological spaces that kernel of sets are open.
1908 ◽
Vol 28 ◽
pp. 249-258
§ 1. THE usual method of proving that a function defined as the limit of a sequence of continuous functions is continuous is by proving that the convergence is uniform. This method may fail owing to the presence of points at which the convergence is non-uniform although the limiting function is continuous. In such a case it would be necessary to apply a further test, e.g. that of Arzelà ("uniform convergence by segments").In some cases the continuity may be proved directly by means of a totally different principle, without reference to modes of convergence at all. It is, in fact, a necessary and sufficient condition for the continuity of a function that it should be possible to express it at the same time as the limit of a monotone ascending and of a monotone descending sequence of continuous functions.
Separable Topological Space of Hereditary
2020 ◽
Vol 7 (1-2) ◽
pp. 68-70
A property of a topological space is termed hereditary ifand only if every subspace of a space with the property also has the property. The purpose of this article is to prove that the topological property of separable space is hereditary. In this paper we determine some topological properties which are hereditary and investigate necessary and sufficient condition functions for sub-spaces to possess properties of sub-spaces which are not in general hereditary.
On Global Inverse Theorems of Szász and Baskakov Operators
1979 ◽
Vol 31 (2) ◽
pp. 255-263 ◽
The Szász and Baskakov approximation operators are given by1.11.2respectively. For continuous functions on [0, ∞) with exponential growth (i.e. ‖ƒ‖A ≡ supx\ƒ(x)e–Ax\ < M) the modulus of continuity is defined by1.3where ƒ ∈ Lip* (∝, A) for some 0 < ∝ ≦ 2 if w2(ƒ, δ, A) ≦ Mδ∝ for all δ < 1. We shall find a necessary and sufficient condition on the rate of convergence of An(ƒ, x) (representing Sn(ƒ, x) or Vn(ƒ, x)) to ƒ(x) for ƒ(x) ∈ Lip* (∝, A). In a recent paper of M. Becker [1] such conditions were found for functions of polynomial growth (where (1 + \x\N)−1 replaced e–Ax in the above). M. Becker explained the difficulties in treating functions of exponential growth.
2005 ◽
Vol 178 ◽
pp. 55-61 ◽
Let α be a nonnegative continuous function on ℝ. In this paper, the author obtains a necessary and sufficient condition for polynomials with gaps to be dense in Cα, where Cα is the weighted Banach space of complex continuous functions ƒ on ℝ with ƒ(t) exp(−α(t)) vanishing at infinity.
Source: https://www.sciencegate.app/document/10.4153/cjm-1972-043-8
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