Subspaces With or Without a Common Complement in Hilbert Spaces

Watch a 54-minute mathematics lecture exploring the properties of subspaces in Hilbert spaces, delivered at the Erwin Schrödinger International Institute's Thematic Programme on "Infinite-dimensional Geometry." Delve into the analysis of pairs of elements in the Grassmann manifold of a separable complex Hilbert space, focusing on subspaces with and without common complements. Learn about the sets D (pairs with common complements) and G (pairs without common complements), their properties as subsets of B(H) x B(H), and discover how D's connected components are parametrized by dimension and codimension. Explore the characteristics of the connected component with infinite dimensional and co-dimensional subspaces, and understand G's nature as a closed C^\infty submanifold. Examine practical examples of these concepts through various pairs in D and connected components of G within function Hilbert spaces.