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Fréchet algebras of power series

H. Garth Dales, Shital R. Patel, Charles J. Read (2010)

Banach Center Publications

We consider Fréchet algebras which are subalgebras of the algebra 𝔉 = ℂ [[X]] of formal power series in one variable and of 𝔉ₙ = ℂ [[X₁,..., Xₙ]] of formal power series in n variables, where n ∈ ℕ. In each case, these algebras are taken with the topology of coordinatewise convergence. We begin with some basic definitions about Fréchet algebras, (F)-algebras, and other topological algebras, and recall some of their properties; we discuss Michael's problem from 1952 on the continuity of characters...

Fréchet differentiability via partial Fréchet differentiability

Luděk Zajíček (2023)

Commentationes Mathematicae Universitatis Carolinae

Let X 1 , ⋯ , X n be Banach spaces and f a real function on X = X 1 × ⋯ × X n . Let A f be the set of all points x ∈ X at which f is partially Fréchet differentiable but is not Fréchet differentiable. Our results imply that if X 1 , ⋯ , X n - 1 are Asplund spaces and f is continuous (respectively Lipschitz) on X , then A f is a first category set (respectively a σ -upper porous set). We also prove that if X , Y are separable Banach spaces and f : X → Y is a Lipschitz mapping, then there exists a σ -upper porous set A ⊂ X such that f is Fréchet differentiable at every...

Fréchet directional differentiability and Fréchet differentiability

John R. Giles, Scott Sciffer (1996)

Commentationes Mathematicae Universitatis Carolinae

Zaj’ıček has recently shown that for a lower semi-continuous real-valued function on an Asplund space, the set of points where the function is Fréchet subdifferentiable but not Fréchet differentiable is first category. We introduce another variant of Fréchet differentiability, called Fréchet directional differentiability, and show that for any real-valued function on a normed linear space, the set of points where the function is Fréchet directionally differentiable but not Fréchet differentiable...

Fréchet interpolation spaces and Grothendieck operator ideals.

Jesús M. Fernández Castillo (1991)

Collectanea Mathematica

Starting with a continuous injection I: X → Y between Banach spaces, we are interested in the Fréchet (non Banach) space obtained as the reduced projective limit of the real interpolation spaces. We study relationships among the pertenence of I to an operator ideal and the pertenence of the given interpolation space to the Grothendieck class generated by that ideal.

Fréchet quotients of spaces of real-analytic functions

P. Domański, L. Frerick, D. Vogt (2003)

Studia Mathematica

We characterize all Fréchet quotients of the space (Ω) of (complex-valued) real-analytic functions on an arbitrary open set Ω ⊆ ℝ d . We also characterize those Fréchet spaces E such that every short exact sequence of the form 0 → E → X → (Ω) → 0 splits.

Fréchet-spaces-valued measures and the AL-property.

S. Okada, W. J. Ricker (2003)

RACSAM

Associated with every vector measure m taking its values in a Fréchet space X is the space L1(m) of all m-integrable functions. It turns out that L1(m) is always a Fréchet lattice. We show that possession of the AL-property for the lattice L1(m) has some remarkable consequences for both the underlying Fréchet space X and the integration operator f → ∫ f dm.

Fredholm determinants

Henry McKean (2011)

Open Mathematics

The article provides with a down to earth exposition of the Fredholm theory with applications to Brownian motion and KdV equation.

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