Comparison of topologies induced by geometric and operator openings of subspaces (Q5916346)

It is known that the operator deviation and the geometric deviation define metrisable topologies on the set of all closed linear subspaces of a Banach space \(X\), the first one of whose is stronger than the second one. For Hilbert spaces these two topologies coincide and in this paper it is proved that, if the above two topologies coincide, then \[ \sup\{p; X\hbox{ has type }p\}=\inf\{q; X\hbox{ has cotype }q\}=2. \]