Given a real separable Hilbert space H, we denote with G(H) the geometry of closed linear subspaces of H.The strong convergence of sequences of subspaces is shown to be a L*-convergence and the weak convergence a L-convergence.The smallest L*-convergence containing the weak convergence is found, and the orthogonal image of the strong convergence, which is also a L*-convergence, is defined.
Si dimostra che ogni funzione multivoca lipschitziana con costante di Lipschitz , definita su un sottoinsieme di uno spazio di Hilbert a valori compatti e convessi in , può essere estesa su tutto ad una funzione multivoca lipschitziana con costante minore di 7 nM. In generale, non esistono invece estensioni aventi la stessa costante di Lipschitz .
We use the Maurey-Rosenthal factorization theorem to obtain a new characterization of multiple 2-summing operators on a product of spaces. This characterization is used to show that multiple s-summing operators on a product of spaces with values in a Hilbert space are characterized by the boundedness of a natural multilinear functional (1 ≤ s ≤ 2). We use these results to show that there exist many natural multiple s-summing operators such that none of the associated linear operators is s-summing...
Two new convergences of closed linear subspaces in the real separable Hilbert space are defined. These are the uniform strong convergence and the simultaneously strong and weak convergence to a single limit. Both convergences are characterized and it is shown that they verify the three axioms of Fréchet.
An example of a finite set of projectors in is exhibited for which no 0-1 measure exists.