Topological classification of chains of linear mappings

We consider systems of linear mappings A1,…,At-1A1,…,At-1 of the formA:U1—A12—A2U3—A3⋯—At-1Utin which U1,…,UtU1,…,Ut are unitary (or Euclidean) spaces and each line is either the arrow →→ or the arrow ←←. Let AA be transformed toB:V1—B1V2—B2V3—B3⋯—Bt-1Vtby a system {φi:Ui→Vi}i=1t of bijections. We say that AA and BB are linearly isomorphic if all φiφi are linear. Considering all UiUi and ViVi as metric spaces, we say that AA and BB are topologically isomorphic if all φiφi and φi-1 are continuous.We prove that AA and BB are topologically isomorphic if and only if they are linearly isomorphic.