Homomorphisms on Function Lattices

Garrido, M. Isabel; Jaramillo, Jesús Á.

doi:10.1007/s00605-002-0011-4

Cited by 46 publications

(53 citation statements)

References 24 publications

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“…In [8] we proved that the family Lip loc (X) is uniformly dense in C(X), the set of all continuous real-valued functions on X. Now, since LS(X) ⊂ Lip loc (X)∩U(X), Theorem 1 gives an analogous result in the frame of uniformly continuous functions.…”

Section: Corollary 1 In Any Metric Space Every Bounded Uniformly Comentioning

confidence: 76%

“…An analogous result was given by Li Pi Su [19] for the algebra Lip loc (X) of locally Lipschitz functions on certain metric spaces X. This was extended by Bustamante and Arrazola in [4] to the class of realcompact metric spaces, and later on in [8] we obtain it for general metric spaces. On the other hand, Weaver in [20] and [21] studies the normed lattice and algebra structures of spaces Lip * (X) and, more generally, of the so-called Lip-spaces.…”

Section: Introductionmentioning

confidence: 73%

“…Simple examples show that this is not true in general for Lip * (X). Nevertheless, a positive result is given also in [8] for the class of complete quasiconvex metric spaces X considering Lip * (X) as a unital vector lattice or as an algebra.…”

Section: Introductionmentioning

confidence: 95%

“…More recently, we have proved in [8] that the Lipschitz structure of an arbitrary complete metric space X is determined by the unital vector lattice structure of Lip (X). Simple examples show that this is not true in general for Lip * (X).…”

Section: Introductionmentioning

confidence: 99%

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Lipschitz-type functions on metric spaces

Garrido

Jaramillo

2008

Journal of Mathematical Analysis and Applications

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In order to find metric spaces X for which the algebra Lip * (X) of bounded Lipschitz functions on X determines the Lipschitz structure of X, we introduce the class of small-determined spaces. We show that this class includes precompact and quasi-convex metric spaces. We obtain several metric characterizations of this property, as well as some other characterizations given in terms of the uniform approximation and the extension of uniformly continuous functions. In particular we show that X is small-determined if and only if every uniformly continuous real function on X can be uniformly approximated by Lipschitz functions.

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Section: Corollary 1 In Any Metric Space Every Bounded Uniformly Comentioning

confidence: 76%

Section: Introductionmentioning

confidence: 73%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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Lipschitz-type functions on metric spaces

Garrido

Jaramillo

2008

Journal of Mathematical Analysis and Applications

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“…A. Jaramillo [8,Theorem 3.10] and N. Weaver [13, Main theorem, Part (d)], and they concern real-valued Lipschitz functions. The first two authors also tackled the matter for vector lattices of real-valued locally Lipschitz functions [8,Theorem 3.16] and real-valued Lipschitz functions in the small [9,Theorem 2], and N. Weaver tackled normal lattices of complex-valued Lipschitz functions vanishing at some fixed point [14,Corollary]. Recently, we have given in [11] the general form of order isomorphisms between spaces of real or complex-valued little Lipschitz functions on compact Hölder metric spaces.…”

Section: 1) T (F )(Y) = T (Y)(f (ϕ(Y)) (F ∈ C(x E) Y ∈ Y )mentioning

confidence: 99%

The uniform separation property and Banach–Stone theorems for lattice-valued Lipschitz functions

Jiménez-Vargas

Campoy

Villegas-Vallecillos

2009

Proc. Amer. Math. Soc.

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RETRACTED: A systematic review on climate change and geo‐environmental factors induced land degradation: Processes, policy‐practice gap and its management strategies

et al. 2022

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Climate change has an effect on ecosystems and people all around the globe and causes land degradation. A long‐term decline in biological output, loss of ecological integrity, and a fall in the economic worth of the land are all signs of “land degradation”, a downbeat trend in land condition. In the red lateritic zone of the tropical area, where “climate change” and “land degradation” are related, they impact one another. On the other hand, due to the emissions of greenhouse gases (GHGs), which slow the rate of carbon absorption, and the extent of various existing land degradation processes, which are worsened in pace and magnitude by “climate change”. The world's expanding population poses a challenge to supplying the rising demand for food in recent decades. The need for additional land for farming, feed, and other forms of production is fuelled by poverty and the destruction of the environment. In tropical regions, “soil erosion” caused by “water and wind, salinization and alkalinity, acidity, losses of organic carbon, nutrient imbalance, pollution/contamination by toxic substances, and soil sealing and capping” all call for immediate action to stop the land degradation process and sustainably restore productivity for environmental security and food security for the world's expanding population. After fulfilling the requirement systematic knowledge about soil, scientific use of natural resources (water, land, minerals, vegetation, air and biodiversity) will present a healthy planet with a healthy ecological system for future generations.

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Homomorphisms on Function Lattices

Cited by 46 publications

References 24 publications

Lipschitz-type functions on metric spaces

Lipschitz-type functions on metric spaces

The uniform separation property and Banach–Stone theorems for lattice-valued Lipschitz functions

RETRACTED: A systematic review on climate change and geo‐environmental factors induced land degradation: Processes, policy‐practice gap and its management strategies

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