In addition to a specific problem Trans-universality for finitely generated groups, I posted also its general form. It should not hurt to provide another special case:
QUESTION: does there exist a Banach space U such that the following three conditions hold:
- (a) every finite-dimensional Banach space is isometrically isomorphic to a subspace of U;
- (b) for every Banach space G that is not finite-dimensional there exists a Banach space H that is generated by G and one additional element x, and such that H is not isometrically isomorphic to any subspace of U;
- (c) U is separable.
(End of Question)
Remark: A harder case would be that of Banach algebras $\ C(X)\ $ over compact spaces $\ X$.
About condition (b):
A ”universal space” peacefully combines contradictory logical tensions. Given a hederitary class $C$ of spaces, one tries to construct a (universal) space $U$ that contains an isomorphic image of every space $\ X\in C.\ $ The condition hederitary may mean something like “the class $C$ contains all closed subsets of members of $C$”, or similar. This roughly means that a universal space is large.
On the other hand, there must be a condition that tells us that the universal space is in a sense small. In the most classical cases (Urysohn, Sierpiński, Tikhonov, Menger, Hurewicz, ...), the universal space is small because it has to belong to the class $C$ itself: a universal space is large among small spaces (and it is one of them).
But in some problems (like the one in this question), the universal space cannot be a member of the original class $C$… Then what? We still need a condition ensuring that the universal space is small in some sense. In different cases, there have been different smallness conditions that depended ad hoc on the particular theory at the hand.
However, in 1961-62 I introduced a regular smallness condition that may apply to a variety of theories, even in cases where $U$ cannot belong to $\ C\ $: we assume that $U$ doesn't contain any abudant space that doesn't belong to C, except for the necessary ones (since, for instance, $U$ itself doesn't belong to $C$). Hence my condition (b).
