Topological classification of chains of linear mappings (Q426079)

Consider the following systems of linear mappings \(A_1,\dots,A_{m-1}\) of the form NEWLINE\[NEWLINE\mathcal A: U_1 \overset {A_1} {\text{---}}U_2\overset{A_2}{\text{ --- }}U_3\overset{A_3}{\text{ --- }} \cdots\overset{A_{t-1}}{ \text{ --- }} U_mNEWLINE\]NEWLINE in which \(U_1,\dots,U_m\) are unitary spaces and each line is either the arrow \(\rightarrow\) or the arrow \(\leftarrow\). Let A be transformed to NEWLINE\[NEWLINE\mathcal B: V_1\overset{B_1}{\text{ --- }}{ V_2}\overset{B_2}{\text{ --- }} V_3\overset{B_3}{\text{ --- }} \cdots\overset{B{t-1}}{\text{ --- }} V_mNEWLINE\]NEWLINE by a system \(\{\Phi_i:U_i\to V_i\}_{i=1}^m\) of bijections. \(A\) and \(B\) are said to be linearly isomorphic if all \(\Phi_i\) are linear. Considering all \(U_i\) and \(V_i\) as metric spaces, then \(A\) and \(B\) are said to be topologically isomorphic if all \(\Phi_i\) and their inverses are continuous. The authors prove that \(A\) and \(B\) are topologically isomorphic if and only if they are linearly isomorphic.