Stabilizing fake Hilbert spaces

Fake lens spaces *

A fake lens space is an orbit space of a free action of a finite cyclic group on a sphere and as such it is a generalization of a classical lens space. The invariants of fake lens spaces described here are their homotopy groups, homology groups, a certain k-invariant, the Reidemeister torsion, the ρ-invariant and certain splitting invariants. We survey the classification of fake lens spaces whi...

Fake Boundary Sets in the Hilbert Cube

For each positive integer n, a o-Z-set B„ in the Hilbert cube 7°° is constructed whose complement s„ = Ix — B„ is not homeomorphic to the pseudointerior j of the Hilbert cube though sn and B„ satisfy: (i) every compact subset of s„ is a Z-set in s„; (ii) s„ X s„ is homeomorphic to s; (iii) Bn admits small maps 7°° -» B„; (iv) s„ satisfies the discrete «-cells property; and (v) Bn is locally (n ...

Arithmetic Fake Projective Spaces and Arithmetic Fake Grassmannians

Hilbert spaces

• Pre-Hilbert spaces: definition • Cauchy-Schwarz-Bunyakowski inequality • Example: spaces ` • Triangle inequality, associated metric, continuity issues • Hilbert spaces, completions, infinite sums • Minimum principle • Orthogonal projections to closed subspaces • Orthogonal complements W⊥ • Riesz-Fischer theorem on linear functionals • Orthonormal sets, separability • Parseval equality, Bessel...

Generalized Frames in Hilbert Spaces