On relativization of convergence in G(H). (Spanish. English summary) Zbl 0561.46014

Given a real separable Hilbert space H, G(H) denotes the geometry of the closed linear subspaces of H, \(S=\{E^{(n)}|\) \(n\in N\}^ a \)sequence of G(H) and [E] the closed linear hull of E. The weak, strong and other convergences in G(H) were defined and characterized in previous papers. Now we study the convergences of sequences \(\{E^{(n)}\cap F| n\in N\}\) when \(\{E^{(n)}\}\) is a convergent sequence and F is a subspace of G(H), and we show that these convergences hold, if this intersection exists. Conversely, given \(\{E^{(n)}\}\) and E, if for each subspace F of G(H) the sequence \(\{E^{(n)}\cap F\}\) converges to \(E\cap F\) in some of the forms defined, the sequence \(\{E^{(n)}\}\) converges according to the same type of convergence.


46C99 Inner product spaces and their generalizations, Hilbert spaces