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Banach Algebras
Published in Hugo D. Junghenn, Principles of Analysis, 2018
Since σ(A)∪{0} $ \sigma (\mathcal{A}) \cup \{0\} $ is weak∗ $ ^* $ compact and since removing a point from a compact space produces a locally compact space, we see that σ(A) $ \sigma (\mathcal{A}) $ is locally compact, proving (a). Thus if σ(A) $ \sigma (\mathcal{A}) $ is not compact, then σ(A)∪{0} $ \sigma (\mathcal{A}) \cup \{0\} $ is the one-point compactification of σ(A) $ \sigma (\mathcal{A}) $ .
Product Measure
Published in Kenneth Kuttler, Modern Analysis, 2017
In the rest of this chapter, unless otherwise stated, (X, S, μ) and (Y, F, λ) will be two σ finite measure spaces. Note that a Radon measure on a σ compact, locally compact space gives an example of a σ finite space. In particular, Lebesgue measure is σ finite.
Locally Starplus-Compactness in L-Topological Spaces
Published in Fuzzy Information and Engineering, 2018
The class of locally compact spaces is far more wider than the class of compact spaces. The locally compact spaces often arise in topology and applications of topology to geometry, analysis and algebra. For example, the study of locally compact abelian group forms the foundation of harmonic analysis. It is well known that every compact space is locally compact but the converse need not be true. For example, the Euclidean space is locally compact but not compact. Topological manifolds share the local properties of Euclidean space and hence are locally compact. A locally compact space can be imbedded in a compact space, which is its compactification. One of the simplest compapctification of a space is the one point compactification, wherein one simply adjoins one new point to the space. The classical example of one point compactification is the embedding of the Gaussian plane of complex numbers into the Riemann sphere. The category of locally compact spaces has been applied in almost every subdiscipline of mathematics and hence it is important to formulate an appropriate version of local compactness in the L-fuzzy setting.