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We give a criterion ensuring that the elementary class of a modular Banach space $E$ (that is, the class of Banach spaces, some ultrapower of which is linearly isometric to an ultrapower of $E$) consists of all direct sums $E{\oplus }_{m}H$, where $H$ is an arbitrary Hilbert space and ${\oplus }_m$ denotes the modular direct sum. Also, we give several families of examples in the class of Nakano direct sums of finite dimensional normed spaces that satisfy this criterion. This yields many new examples of uncountably categorical Banach...

This is a contribution to the theory of topological vector spaces within the framework of the alternative set theory. Using indiscernibles we will show that every infinite set $uS\subseteq G$ in a biequivalence vector space $\langle W,M,G\rangle$, such that $x-y\notin M$ for distinct $x,y\in u$, contains an infinite independent subset. Consequently, a class $X\subseteq G$ is dimensionally compact iff the $\pi$-equivalence ${\doteq }_M$ is compact on $X$. This solves a problem from the paper [NPZ 1992] by J. Náter, P. Pulmann and the second author.