We say that a Banach space X is the unique predual of its dual (more precisely the unique isometric predual of its dual) in case it is isometric to any Banach space whose dual is isometric to the dual of X. (We say that two Banach spaces Y and Z are isomorphic if there is a bounded linear bijective operator T : Y → Z with bounded inverse T; if moreover ‖T (y)‖ = ‖y‖ for all y ∈ Y we say that Y ...

In this article we study holomorphic isometries of the Poincaré disk into bounded symmetric domains. Earlier we solved the problem of analytic continuation of germs of holomorphic maps between bounded domains which are isometries up to normalizing constants with respect to the Bergman metric, showing in particular that the graph V0 of any germ of holomorphic isometry of the Poincaré disk ∆ into...

A subgraph $H$ of a graph $G$ is isometric if the distances between vertices in coincide with corresponding $G$. We show that for any integer $n\ge 1$, there on $3^{n+O(\log^2 n)}$ contains copies all $n$-vertex graphs. Our main tool new type distance labelling scheme, whose study might be independent interest.