Isometric embeddings of separable Banach spaces into $({\ell }^{\infty }\setminus c)\cup \{0\}$

Abstract

The classical Banach--Mazur theorem asserts that every separable Banach space admits an isometric embedding into $C[0,1]$. It is also well known that every separable Banach space embeds isometrically into ${\ell }^{\infty }$. We show that such an embedding can be chosen so that its image intersects $c$ only at the origin. Moreover, we prove that any finite- or countable-dimensional, or more generally separable, subspace of $({\ell }^{\infty }\setminus c)\cup \{0\}$ can be extended to a subspace containing an isometric copy of an arbitrary separable Banach space, while still avoiding $c$. We further establish that this extension property also holds for every subspace $D\subset {\ell }^{\infty }$ with $D\cap c=\{0\}$ and separable image in the quotient ${\ell }^{\infty }/c$.

10 pages, no figures. Keywords: isometric embeddings; separable Banach spaces; convergent sequences; lineability; spaceability