Person:
Gómez Gil, Javier

First Name

Javier

Last Name

Gómez Gil

Affiliation

Universidad Complutense de Madrid

Faculty / Institute

Ciencias Matemáticas

Department

Análisis Matemático Matemática Aplicada

Area

Análisis Matemático

Identifiers

Full item page

Search Results

Now showing 1 - 10 of 15

Rolle’s Theorem and Negligibility of Points in Infinite Dimensional Banach Spaces
(Elsevier, 1997-09-15) Azagra Rueda, Daniel; Gómez Gil, Javier; Jaramillo Aguado, Jesús Ángel
In this note we prove that if a differentiable function oscillates between y« and « on the boundary of the unit ball then there exists a point in the interior of the ball in which the differential of the function has norm equal or less than« . This kind of approximate Rolle’s theorem is interesting because an exact Rolle’s theorem does not hold in many infinite dimensional Banach spaces. A characterization of those spaces in which Rolle’s theorem does not hold is given within a large class of Banach spaces. This question is closely related to the existence of C1 diffeomorphisms between a Banach space X and X _ _04 which are the identity out of a ball, and we prove that such diffeomorphisms exist for every C1 smooth Banach space which can be linearly injected into a Banach space whose dual norm is locally uniformly rotund (LUR).
Superdifferential Analysis of the Takagi-Van Der Waerden Functions
(Springer Nature, 2021-11-22) Ferrera Cuesta, Juan; Gómez Gil, Javier; Llorente Márquez, Jesús
In this work we completely describe the superdifferential of the Takagi-Van der Waerden functions and, as a consequence, the local maxima of these functions are characterized. Regarding the set of points where the superdifferential is not empty, we calculate its Hausdorff dimension as well as its corresponding Hausdorff measure. To do so, for any even integer greater than or equal to two we determine the 1/2-dimensional Hausdorff measure of the set of points where Takagi-Van der Waerden functions attain their global maximum.
Polynomial approximation of weakly differentiable functions on banach spaces
(Royal Irish Academy, 1982) Gómez Gil, Javier; Llavona, José G.
A characterization of dense polynomial algebras is obtained for a certain class of p-continously differentiable Fréchet functions between real Banach spaces. In order to study this result a new approximation property in Banach spaces, the bounded weak approximation property, is introduced and studied.
Whitney's Theorem: A nonsmooth version
(Elsevier, 2015-11-01) Ferrera Cuesta, Juan; Gómez Gil, Javier
In this paper we solve the problem of extending continuous functions with nonempty subdifferential at every point of a closed subset A of R-n to functions with the same property defined in the whole Rn, keeping the property of outer semicontinuity of the subdifferential, which is a set-valued function. The proof is constructive, and gives us a wide range of possible extensions.
Espacios de funciones debilmente diferenciables
(Universidad Complutense de Madrid, 2015) Gómez Gil, Javier; González Llavona, Luis
On completion of spaces of weakly continuous functions.
(LONDON MATH SOC, 1983) Ferrera Cuesta, Juan; Gómez Gil, Javier; Llavona, José G.
Let E and F be two Banach spaces and let A be a nonempty subset of E . A mapping f:A→F is said to be weakly continuous if it is continuous when A has the relative weak topology and F has the topology of its norm. Let A={E} , B= {A⊂E:A is bounded} and C= {A⊂E:A is weakly compact}. Then C w (E;F) , C wb (E;F) and C wk (E;F) are the spaces of all mappings f:E→F whose restrictions to subsets A⊂E belonging to A , B and C , respectively, are weakly continuous. Clearly, C w (E;F)⊂C wb (E;F)⊂C wk (E;F) , and they are all endowed with the topology of uniform convergence on weakly compact subsets of E . The authors show that C wk (E;F) is the completion of C w (E;F) . They also show that, when E has no subspace isomorphic to l 1 , then C wb (E;F)=C wk (E;F) . When E has the Dunford-Pettis property and contains a subspace isomorphic to l 1 , the authors prove that C wb (E;F) is a proper subspace of C wk (E;F) . The same conclusion holds when E is a Banach space that contains a subspace isomorphic to l ∞ .
Nonsmooth Morse–Sard theorems
(Elsevier, 2017) Azagra Rueda, Daniel; Ferrera Cuesta, Juan; Gómez Gil, Javier
We prove that every function f:Rn→R satisfies that the image of the set of critical points at which the function f has Taylor expansions of order n−1 and non-empty subdifferentials of order n is a Lebesgue-null set. As a by-product of our proof, for the proximal subdifferential ∂P, we see that for every lower semicontinuous function f:R2→R the set f({x∈R2:0∈∂Pf(x)}) is L1-null.
Interpolation by Weakly Differentiable Functions on Banach-Spaces
(Academic Press, 1994-03) Gómez Gil, Javier; Jaramillo Aguado, Jesús Ángel
Let (a(n)) be a weakly null Schauder basis of a Banach space E, and let (lambda(n)) be a convergent sequence of real numbers. We study the problem of finding an m-times weakly uniformly differentiable function f on E such that f(a(n)) = lambda(n). We prove that this problem has always a solution for m = 1. In some cases we find a solution for m = infinity, for instance when E is super-reflexive or when (a(n)) is a symmetric basis and E does not contain a copy of c0. In these cases we obtain as a consequence the nonreflexivity of the space of infinitely weakly uniformly differentiable functions on E.
Homomorphisms on some function algebras
(Universidad de Extremadura, Departamento de Matemáticas, 1992) Garrido, M. Isabel; Gómez Gil, Javier; Jaramillo Aguado, Jesús Ángel
For an algebra A of continuous real-valued functions on a topological space X, the question of whether every algebra homomorphism is a point evaluation for a point in X is considered. A variety of results are provided, such as the following. Let X be completely regular and A⊂C(X) a subalgebra with unit which is closed under bounded inversion and separates points and closed sets. Then every homomorphism is a point evaluation for a point in X if and only if, for each point z in the Stone-Čech compactification of X and not in X, there exists a function in A whose extension to z is infinite. Examples are considered and further results for the case of functions on a Banach space are discussed
Multiplicatively functionals on function álgebras
(Universidad Complutense de Madrid, 1988) Gómez Gil, Javier; Llavona, José G.