Topologically independent sets in topological groups and vector spaces
Jan Spěvák
TL;DR
The paper develops topological analogues of independence for abelian groups and vector spaces, introducing topological independence and topologically linear independence and tying them to Tychonoff direct sums via the Kalton map. It generalizes known results from precompact to arbitrary subgroups of $({\mathbb C}^\times)^I$, proving that a subset $A$ is topologically independent iff $\langle A\rangle$ is the direct sum $\bigoplus_{a\in A}\langle a\rangle$. In weak topologies, topological independence and topological linear independence coincide, with further characterizations in terms of minimality and equicontinuity; in normed spaces, topological linear independence is equivalent to uniform minimality and bounded biorthogonal systems, linking to Markushevich bases and $\ell^1$-representations. These results unify structural understandings of independence-like sets across groups and vector spaces and reveal how topology (weak vs normed) shapes these notions and their applications to bases and representations.
Abstract
We study topological versions of an independent set in an abelian group and a linearly independent set in a vector space, a {\em topologically independent set} in a topological group and a {\em topologically linearly independent set} in a topological vector space. These counterparts of their algebraic versions are defined analogously and possess similar properties. Let $\C^\times$ be the multiplicative group of the field of complex numbers with its usual topology. We prove that a subset $A$ of an arbitrary Tychonoff power of $\C^\times$ is topologically independent if and only if the topological subgroup $\hull{A}$ that it generates is the Tychonoff direct sum $\bigoplus_{a\in A}\hull{a}$. This theorem substantially generalizes an earlier result of the author, who has proved this for Abelian precompact groups. Further, we show that topologically independent and topologically linearly independent sets coincide in vector spaces with weak topologies, although they are different in general. We characterize topologically linearly independent sets in vector spaces with weak topologies and normed spaces. In a weak topology, a set $A$ is topologically linearly independent if and only if its linear span is the Tychonoff direct sum $\R^{(A)}$. In normed spaces $A$ is topologically linearly independent if and only if it is uniformly minimal. Thus, from the point of view of topological linear independence, the Tychonoff direct sums $\R^{(A)}$ and (linear spans of) uniformly minimal sets, which are closely related to bounded biorthogonal systems, are of the same essence.