Applying coordinate products to the topological identification of normed spaces

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.