An inverse function theorem in Fréchet-Spaces

An inverse function theorem in Fréchet spaces

I present an inverse function theorem for differentiable maps between Fréchet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue’s dominated convergence theorem and Ekeland’s variational principle. As a consequence, the assumptions are substantially weakened: th...

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...

On Consistency in Parameter Spaces of Expanding Dimension: an Application of the Inverse Function Theorem

Foutz (1977) uses the Inverse Function Theorem to prove the existence of a unique and consistent solution to the likelihood equations. This note extends his results in three useful directions. The first is to remark that with minor modification the same proof may be used to show that the solution to the likelihood equations converges asymptotically to the least-false parameter (Hjort (1986, 199...

Rybakov ’ S Theorem in Fréchet Spaces and Completeness of L 1 - Spaces

We provide a simple and direct proof of the completeness of the L1-space of any vector measure taking its values in the class of Fréchet spaces which do not contain a copy of the sequence space !. 1991 Mathematics subject classification (Amer. Math. Soc.): primary 28B05. Bartle, Dunford and Schwartz [1] (respectively Kluvánek and Knowles [6], and Lewis [7]) developed integration theories in Ban...

On the Inverse Function Theorem