On Banach spaces of continuous functions on finite products of separable compact lines
Volume 251 / 2020
Studia Mathematica 251 (2020), 247-275
MSC: Primary 46E15, 46B03.
DOI: 10.4064/sm180507-3-1
Published online: 25 October 2019
Abstract
We study properties of the Banach spaces $C(L,X)$ of all continuous functions from a finite product $L$ of compact lines into a Banach space $X$. We show that if $L_1,\dots , L_k$, $K_1,\dots ,K_n$ are nonmetrizable separable compact lines and $X,Y$ are separable Banach spaces, then (1) the space $C(L_1\times \dots \times L_k, X)$ is not isomorphic to any subspace of $C(K_1\times \dots \times K_n,Y)$ whenever $k \gt n$, (2) there is no continuous linear surjection from $C(L_1\times \dots \times L_k, X)$ onto $C(K_1\times \dots \times K_n,Y)$ whenever $k \lt n$.
