Differential calculus in Fréchet spaces

A New Differential Calculus on Noncommutative Spaces

We develop a GL qp (2) invariant differential calculus on a two-dimensional noncommutative quantum space. Here the coordinate space for the exterior quantum plane is spanned by the differentials that are commutative (bosonic) in nature.

Bernstein's Lethargy Theorem in Fréchet spaces

In this paper we consider Bernstein’s Lethargy Theorem (BLT) in the context of Fréchet spaces. Let X be an infinite-dimensional Fréchet space and let V = {Vn} be a nested sequence of subspaces of X such that Vn ⊆ Vn+1 for any n ∈ N and X = ⋃∞ n=1 Vn. Let en be a decreasing sequence of positive numbers tending to 0. Under an additional natural condition on sup{dist(x, Vn)}, we prove that there e...

Fréchet on the Calculus of Probability

Restrictive metric regularity and generalized differential calculus in Banach spaces

We consider nonlinear mappings f :X → Y between Banach spaces and study the notion of restrictive metric regularity of f around some point x̄, that is, metric regularity of f from X into themetric space E = f(X). Some sufficient as well as necessary and sufficient conditions for restrictive metric regularity are obtained, which particularly include an extension of the classical Lyusternik-Graves...

Weak symplectic forms and differential calculus in Banach spaces

1Jerrold E. Marsden and Tudor S. Ratiu, Introduction to Mechanics and Symmetry, second ed., Chapter 2. 2Serge Lang, Differential and Riemannian Manifolds, p. 150, Theorem 8.1; Mircea Puta, Hamiltonian Mechanical Systems and Geometric Quantization, p. 12, Theorem 1.3.1. 3Andreas Kriegl and Peter W. Michor, The Convenient Setting of Global Analysis, p. 522, §48; Peter W. Michor, Some geometric ev...