On the topological linear spaces whose subspace lattices are modular (Q1202750)

Let \(X\) be a Hausdorff topological linear space and \(L(X)\) be the lattice of all closed subspaces of \(X\). The modularity conditions for the lattice \(L(X)\) which garanties that only the finite dimensional subsets of \(X\) are bounded in \(X\) are derived. The conditions are based on not containing an isomorphic copy of the space \(\omega\) of all sequences endowed with the product topology. Some applications of the obtained results in the theory of topological linear spaces and also outside this theory for example on quantum axiomatics are suggested.