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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Applying coordinate products to the topological identification of normed spaces

Authors: Robert Cauty and Tadeusz Dobrowolski
Journal: Trans. Amer. Math. Soc. 337 (1993), 625-649
MSC: Primary 57N17; Secondary 46B99
MathSciNet review: 1210952
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Abstract: Using the $ {l^2}$-products we find pre-Hilbert spaces that are absorbing sets for all Borelian classes of order $ \alpha \geq 1$. We also show that the following spaces are homeomorphic to $ \Sigma^\infty$, the countable product of the space $ \Sigma = \{(x_n) \in R^\infty: (x_n)$ is bounded}:

(1) every coordinate product $ \prod_C H_n$ of normed spaces $ H_n$ in the sense of a Banach space $ C$, where each $ H_n$ is an absolute $ F_{\sigma\delta}$-set and infinitely many of the $ H_n$'s are $ {Z_\sigma }$-spaces,

(2) every function space $ \tilde{L}^p = \cap_{p\prime <p}L^{p\prime}$ with the $ {L^q}$-topology, $ 0<q<p \leq \infty$,

(3) every sequence space $ {\tilde l^p} = { \cap _{p < p\prime}}{l^{p\prime}}$ with the $ l^q$-topology, $ 0 \leq p < q < \infty$.

We also note that each additive and multiplicative Borelian class of order $ \alpha \geq 2$, each projective class, and the class of nonprojective spaces contain uncountably many topologically different pre-Hilbert spaces which are $ Z_\sigma$-spaces.

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Keywords: Coordinate product, pre-Hilbert space, strong $ (\mathcal{K},\mathcal{L})$-universality, absorbing set, absolute $ F_{\sigma\delta}$-set
Article copyright: © Copyright 1993 American Mathematical Society