One property of families of imbedded Banach spaces (Q1190058)

Let \(\{E_\alpha: \alpha\in[0, 1]\}\) be a family of imbedded Banach spaces, i.e., for each pair of numbers \(\alpha<\beta\) \((\alpha,\beta\in [0,1])\) there is a one-to-one bounded linear mapping (in what follows, imbedding) \(T_{\beta\alpha}: E_\beta\to E_\alpha\), and the following conditions are satisfied: 1) \(\text{cl }T_{\beta\alpha} E_\beta= E_\alpha\), i.e., \(T_{\beta\alpha}\) is a dense imbedding; 2) \(T_{\alpha\gamma} T_{\beta\alpha}= T_{\beta\gamma}\), \(0\leq \gamma< \alpha<\beta\leq 1\); 3) \(T_{\beta\alpha}^{-1}\) is an unbounded mapping. Under these conditions, the author proves two theorems including Theorem 2. Let \(\{E_\alpha: \alpha\in [0,1]\}\) be a family of imbedded separable Banach spaces. Then, for every \(\alpha\in [0,1]\), the space \(E_\alpha\) contains a subspace \(G_\alpha\) with basis \(\{g_i\}\) such that 1) \(G_\alpha\cap \bigcup_{\beta> \alpha} T_{\beta\alpha} E_\beta= 0\); 2) for all \(\gamma< \alpha\) the system \(\{T_{\alpha\gamma} g_i\}_{i= 1}^\infty\) is an \(M\)-basis of the space \(E_\gamma\); in particular, the image \(T_{\alpha\gamma} G_\alpha\) is dense in \(E_\gamma\).