For a locally compact Hausdorff space K and a Banach space X let C₀(K, X) denote the space of all continuous functions f:K → X which vanish at infinity, equipped with the supremum norm. If X is the scalar field, we denote C₀(K, X) simply by C₀(K). We prove that for locally compact Hausdorff spaces K and L and for a Banach space X containing no copy of c₀, if there is an isomorphic embedding of C₀(K) into C₀(L,X), then either K is finite or |K| ≤ |L|. As a consequence, if there is an isomorphic embedding of C₀(K) into C₀(L,X) where X contains no copy of c₀ and L is scattered, then K must be scattered.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm7857-3-2016,
author = {Leandro Candido},
title = {On embeddings of C0(K) spaces into C0(L,X) spaces},
journal = {Studia Mathematica},
volume = {233},
year = {2016},
pages = {1-6},
zbl = {06575019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm7857-3-2016}
}
Leandro Candido. On embeddings of C₀(K) spaces into C₀(L,X) spaces. Studia Mathematica, Tome 233 (2016) pp. 1-6. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm7857-3-2016/