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Generators for algebras dense in L p -spaces

Alexander J. Izzo, Bo Li (2013)

Studia Mathematica

For various L p -spaces (1 ≤ p < ∞) we investigate the minimum number of complex-valued functions needed to generate an algebra dense in the space. The results depend crucially on the regularity imposed on the generators. For μ a positive regular Borel measure on a compact metric space there always exists a single bounded measurable function that generates an algebra dense in L p ( μ ) . For M a Riemannian manifold-with-boundary of finite volume there always exists a single continuous function that generates...

Local/global uniform approximation of real-valued continuous functions

Anthony W. Hager (2011)

Commentationes Mathematicae Universitatis Carolinae

For a Tychonoff space X , C ( X ) is the lattice-ordered group ( l -group) of real-valued continuous functions on X , and C * ( X ) is the sub- l -group of bounded functions. A property that X might have is (AP) whenever G is a divisible sub- l -group of C * ( X ) , containing the constant function 1, and separating points from closed sets in X , then any function in C ( X ) can be approximated uniformly over X by functions which are locally in G . The vector lattice version of the Stone-Weierstrass Theorem is more-or-less equivalent...

The Banach–Mazur game and σ-porosity

Miroslav Zelený (1996)

Fundamenta Mathematicae

It is well known that the sets of the first category in a metric space can be described using the so-called Banach-Mazur game. We will show that if we change the rules of the Banach-Mazur game (by forcing the second player to choose large balls) then we can describe sets which can be covered by countably many closed uniformly porous sets. A characterization of σ-very porous sets and a sufficient condition for σ-porosity are also given in the terminology of games.

The structure of the σ -ideal of σ -porous sets

Miroslav Zelený, Jan Pelant (2004)

Commentationes Mathematicae Universitatis Carolinae

We show a general method of construction of non- σ -porous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each non- σ -porous Suslin subset of a topologically complete metric space contains a non- σ -porous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a non- σ -porous element. Namely, if we denote the space of all compact subsets of a compact metric space E with the Vietoris topology...

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