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

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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 $\ell^\infty/c$.