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On vectorial inner product spaces

João de Deus Marques (2000)

Czechoslovak Mathematical Journal

Let E be a real linear space. A vectorial inner product is a mapping from E × E into a real ordered vector space Y with the properties of a usual inner product. Here we consider Y to be a ℬ -regular Yosida space, that is a Dedekind complete Yosida space such that ⋂ J ∈ ℬ J = { 0 } , where ℬ is the set of all hypermaximal bands in Y . In Theorem 2.1.1 we assert that any ℬ -regular Yosida space is Riesz isomorphic to the space B ( A ) of all bounded real-valued mappings on a certain set A . Next we prove Bessel Inequality and Parseval...

On weak drop property and quasi-weak drop property

J. H. Qiu (2003)

Studia Mathematica

Every weakly sequentially compact convex set in a locally convex space has the weak drop property and every weakly compact convex set has the quasi-weak drop property. An example shows that the quasi-weak drop property is strictly weaker than the weak drop property for closed bounded convex sets in locally convex spaces (even when the spaces are quasi-complete). For closed bounded convex subsets of quasi-complete locally convex spaces, the quasi-weak drop property is equivalent to weak compactness....

On weighted inductive limits of non-Archimedean spaces of continuous functions

A. K. Katsaras, V. Benekas (2000)

Bollettino dell'Unione Matematica Italiana

Si studiano alcune proprietà di un certo limite induttivo di spazi non-archimedei di funzioni continue. In particolare, si esamina la completezza di questo limite induttivo e si indaga il problema di quando lo spazio coincide con il proprio inviluppo proiettivo.

Open partial isometries and positivity in operator spaces

David P. Blecher, Matthew Neal (2007)

Studia Mathematica

We first study positivity in C*-modules using tripotents ( = partial isometries) which are what we call open. This is then used to study ordered operator spaces via an "ordered noncommutative Shilov boundary" which we introduce. This boundary satisfies the usual universal diagram/property of the noncommutative Shilov boundary, but with all the arrows completely positive. Because of their independent interest, we also systematically study open tripotents and their properties.

Open Subsets of LF-spaces

Kotaro Mine, Katsuro Sakai (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

Let F = ind lim Fₙ be an infinite-dimensional LF-space with density dens F = τ ( ≥ ℵ ₀) such that some Fₙ is infinite-dimensional and dens Fₙ = τ. It is proved that every open subset of F is homeomorphic to the product of an ℓ₂(τ)-manifold and ℝ ∞ = i n d l i m ℝ ⁿ (hence the product of an open subset of ℓ₂(τ) and ℝ ∞ ). As a consequence, any two open sets in F are homeomorphic if they have the same homotopy type.

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