The area of research of this paper goes back to a 1930 result of H. Auerbach showing that a scalar series is (absolutely) convergent if all its zero-density subseries converge. A series ${\sum }_n{x}_n$ in a topological vector space X is called ℒ-convergent if each of its lacunary subseries ${\sum }_k{x}_{n_k}$ (i.e. those with $n_{k+1}-n_k\to \infty$) converges. The space X is said to have the Lacunary Convergence Property, or LCP, if every ℒ-convergent series in X is convergent; in fact, it is then subseries convergent. The Zero-Density Convergence...