Complemented subspaces of products of Banach spaces

Chigogidze, Alex

doi:10.1090/s0002-9939-01-06054-3

Cited by 1 publication

(1 citation statement)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Injective Frechét spaces include injective Banach spaces (see [31; Lemma 0]) and hence their countable products, and these appear to be all known examples (see [25] and [30]).…”

Section: Proof (A) the Implicationsmentioning

confidence: 99%

Metrizable uniform spaces

Melikhov¹

2011

Preprint

View full text Add to dashboard Cite

Three themes of general topology: quotient spaces; absolute retracts; and inverse limits -are reapproached here in the setting of metrizable uniform spaces, with an eye to applications in geometric and algebraic topology. The results include:a uniformly continuous partial map of metric spaces, where A is closed in X, we show that the adjunction space X ∪ f Y with the quotient uniformity (hence also with the topology thereof) is metrizable, by an explicit metric. This yields natural constructions of cone, join and mapping cylinder in the category of metrizable uniform spaces, which we show to coincide with those based on subspace (of a normed linear space); on product (with a cone); and on the isotropy of the l 2 metric.2) We revisit Isbell's theory of uniform ANRs, as refined by Garg and Nhu in the metrizable case. The iterated loop spaces Ω n P of a pointed compact polyhedron P are shown to be uniform ANRs. Four characterizations of uniform ANRs among metrizable uniform spaces X are given: (i) the completion of X is a uniform ANR, and the remainder is uniformly a Z-set in the completion; (ii) X is uniformly locally contractible and satisfies the Hahn approximation property; (iii) X is uniformly ε-homotopy dominated by a uniform ANR for each ε > 0; (iv) X is an inverse limit of uniform ANRs with "nearly splitting" bonding maps. * := C * ∧D X is a uniform cover of X star-refining C and D ′ * := D * ∧C Y is a uniform cover of Y star-refining D. In addition, C ′ * ∩ A = F * = D ′ * ∩ A. Next, let C * * be a uniform cover of X star-refining C ′ * and let D * * be a uniform cover of Y star-refining D ′ * . Let F * * = C * * ∧ D * * , and define C ′ * * and D ′ * * similarly to the above. Then C ′ * * is a uniform cover of X star-refining C ′ * and D ′ * * is a uniform cover of Y star-refining D ′ * ; in addition, C ′

show abstract

“…Injective Frechét spaces include injective Banach spaces (see [31; Lemma 0]) and hence their countable products, and these appear to be all known examples (see [25] and [30]).…”

Section: Proof (A) the Implicationsmentioning

confidence: 99%

Metrizable uniform spaces

Melikhov¹

2011

Preprint

View full text Add to dashboard Cite

show abstract

Complemented subspaces of products of Banach spaces

Abstract: Abstract. We show that complemented subspaces of uncountable products of Banach spaces are products of complemented subspaces of countable subproducts.

Cited by 1 publication

References 8 publications

Metrizable uniform spaces

Metrizable uniform spaces

Contact Info

Product

Resources

About