A generalization of the Flow-box Theorem is proven. The assumption of a C vector field f is relaxed to the condition that f be locally Lipschitz. The theorem holds in any Banach space.

We prove analogs of Thom’s transversality theorem and Whitney’s theorem on immersions for pseudo-holomorphic discs. We also prove that pseudo-holomorphic discs form a Banach manifold. MSC: 32H02, 53C15.

We study model theoretical stability for structures from functional analysis. We prove a functional-analytic version of the Finite Equivalence Relation Theorem. We also the Stability Spectrum Theorem for Banach space structures.

There is a subtle difference as far as the invariant subspace problem is concerned for operators acting on real Banach spaces and operators acting on complex Banach spaces. For instance, the classical hyperinvariant subspace theorem of V. I. Lomonosov [10] while true for complex Banach spaces is false for real Banach spaces. When one starts with a bounded operator on a real Banach space and the...