Hilbert spaces with generic predicates

Hilbert spaces with generic groups of automorphisms

Let G be a countable group. We proof that there is a model companion for the approximate theory of a Hilbert space with a group G of automorphisms. We show that G is amenable if and only if the structure induced by countable copies of the regular representation of G is existentially

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...

On perturbations of Hilbert spaces and probability algebras with a generic automorphism

We prove that IHSA, the theory of infinite dimensional Hilbert spaces equipped with a generic automorphism, is א0-stable up to perturbation of the automorphism, and admits prime models up to perturbation over any set. Similarly, APrA, the theory of atomless probability algebras equipped with a generic automorphism is א0-stable up to perturbation. However, not allowing perturbation it is not eve...

Generalized Frames in Hilbert Spaces

Banach Spaces and Hilbert Spaces

A sequence {vj} is said to be Cauchy if for each > 0, there exists a natural number N such that ‖vj−vk‖ < for all j, k ≥ N . Every convergent sequence is Cauchy, but there are many examples of normed linear spaces V for which there exists non-convergent Cauchy sequences. One such example is the set of rational numbers Q. The sequence (1.4, 1.41, 1.414, . . . ) converges to √ 2 which is not a ra...