Let X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover $$ \mathcal{U} $$ of X there is a sequence of maps (f n: X → X)nεgw such that each f n is $$ \mathcal{U} $$-near to the identity map of X and the family {f n(X)}n∈ω is locally finite in X. Also we show that a metrizable space X of density dens(X) < $$ \mathfrak{d} $$ is a Hilbert manifold if X has gw-LFAP and each closed subset A ⊂ X of density dens(A) < dens(X) is a Z ∞-set in X.
2
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
We discuss Fredholm pairs of subspaces and associated Grassmannians in a Hilbert space. Relations between several existing definitions of Fredholm pairs are established as well as some basic geometric properties of the Kato Grassmannian. It is also shown that the so-called restricted Grassmannian can be endowed with a natural Fredholm structure making it into a Fredholm Hilbert manifold.
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.