Differentiable functions on certain Banach spaces (original) (raw)
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A Characterization of C -Smooth Banach Spaces
Bulletin of the London Mathematical Society, 1990
We show that if A" is a Banach space, A'does not contain a subspace isomorphic to c o (N) and there is on X a nonzero real-valued C^-smooth function with bounded support if and only if there is a polynomial on X which vanishes at the origin and with values greater than 1 on the unit sphere. In this paper, we call a bump function on a Banach space X a nonzero real-valued function on X with bounded support, and we shall say that X is C°°-smooth if there is on A' a C°°-smooth bump function. We consider here Banach spaces X which have no subspace isomorphic to c o {N) and, for these spaces, we write X :p c 0. The geometry of C^-smooth Banach spaces have been investigated in [3] where it is shown that if A'is a C°°-smooth Banach space, then either X contains an isomorphic copy of c o (f^J) or X contains an isomorphic copy of l v (N) for some even integer p ^ 2. This result is obtained via a study of cotype of C°°-smooth Banach spaces which do not contain an isomorphic copy of c Q. Our aim here is to give a complete characterization of C°°-smooth Banach spaces which have no subspaces isomorphic to c 0. This is done in Theorem 2. This characterization relies on Theorem 1 which shows that if X is a C*-smooth Banach space of cotype q with q < k, then X is automatically a C°°-smooth Banach space. We then show in Corollary 1 that Theorem 1 can be viewed as a generalization of a result of V. Z. Meshkov [7]. Let us recall the definitions that we shall need in this paper. DEFINITION 1. A Banach space X is of cotype q if there is a constant C such that Mx x ,x 2 ,...,x n eX LWI 9^c ||£e (Jr where *\ 2 *r±l We refer the reader to [6] for the definition of type and for basic results on these notions. DEFINITION 2. Let p be a real number greater than 1 and k be the greatest integer strictly less than p. A function / from X into U is //"-smooth if / is k times continuously differentiable and, for every xeX there exists 8, M > 0 such that: \\y-x\\ ^ 8 and ||Z-JC|| ^ 8 imply \[f m (y)-f {k) (z)|| ^ M\\y-z\\ vk .
A generic view on the theorems of Brouwer and Schauder
Mathematische Zeitschrift, 1993
Several generic fixed point theorems have been established by De Blasi [5, 6], Myjak [9], De Blasi and Myjak [7], Myjak and Sampalmieri [101, Butler [4], Dominguez Benavides [8], Vidossich [121, to quote just a few. However it seems that the well-known and important theorem of Schauder [11] and its finitedimensional version, Brouwer's theorem [31, have not yet been generically investigated. We now fill this gap. For a survey of generic results on convex bodies in Euclidean spaces, see [15]. A large set of fixed points Let E be a Banach space and consider a compact convex set K c E with more than one point. By the Schauder theorem, every continuous function f: K ~ K has at least one fixed point. We shall show here that, in the sense of Baire categories, most continuous functions have uncountably many fixed points. More precisely, the set of all their fixed points is homeomorphic to the Cantor set. This sharply contrasts with the generic finiteness of the fixed point set found by De Blasi [6] in another space of functions and with the behaviour of the non-expansive mappings which have, generically, a single fixed point ([12], see also [9, p. 29]). Of course, the space ~g(K) of all continuous functions f: K-~ K, equipped with the metric of uniform convergence, is complete, hence a Baire space. "Most" means "all, except those in a first category set" (so, a property is generic if it is shared by most elements). Forf~ Cg(K), let Ff be the set of all fixed points off For x ~ E, B(x, r) denotes the open ball of centre x and radius r. The proof of Theorem A below will make use of the following recent result.
A Theorem of Type “Partition of Unity” with
Mathematical Reports, 2022
In the first part of the paper, we present a theorem of type "partition of unity", and three of its consequences, i.e., for algebras, linear subspaces and convex cones. In the second part, some theorems of localizability (or density) in a weighted space are presented. We mention that the weighted spaces are classes of continuous scalar functions on a locally compact space (for example, the space of function with compact support, the space of bounded functions, the space of functions vanishing at infinity, the space of functions rapidly decreasing at infinity).
A generalization of Riesz's uniqueness theorem
arXiv: Complex Variables, 2005
There have been, over the last 8 years, a number of far reaching extensions of the famous original F. and M. Riesz's uniqueness theorem that states that if a bounded analytic function in the unit disc of the complex plane BbbC\Bbb CBbbC has the same radial limit in a set of positive Lebesgue measure on its boundary, then the function has to be constant. First Beurling [B], considering the case of non-constant meromorphic functions mapping the unit disc on a Riemann surface of finite spherical area, was able to prove that if such a function showed an appropriate behavior in the neighborhood of the limit value where the function maps a set on the boundary of the unit disc, then those sets have logarithmic capacity zero. The author of the present note, in [V], was able to weaken Beurling's condition on the limit value. Those results where quite restrictive in a two folded way, namely, they were in dimension n=2n=2n=2 and the regularity requirements on the treated functions were quite str...
Transactions of the American Mathematical Society, 1991
We investigate the classes of Banach spaces where analogues of the classical Hardy inequality and the Paley gap theorem hold for vector-valued functions. We show that the vector-valued Paley theorem is valid for a large class of Banach spaces (necessarily of cotype 2 2 ) which includes all Banach lattices of cotype 2 2 , all Banach spaces whose dual is of type 2 2 and also the preduals of C ∗ {C^ * } -algebras. For the trace class S 1 {S_1} and the dual of the algebra of all bounded operators on a Hilbert space a stronger result holds; namely, the vector-valued analogue of the Fefferman theorem on multipliers from H 1 {H^1} into l 1 {l^1} ; in particular for the latter spaces the vector-valued Hardy inequality holds. This inequality is also true for every Banach space of type > 1 > 1 (Bourgain).
On the range of the derivatives of a smooth function between Banach spaces
Mathematical Proceedings of the Cambridge Philosophical Society, 2003
We study the size of the range of the derivatives of a smooth function between Banach spaces. We establish conditions on a pair of Banach spaces X and Y to ensure the existence of a C p smooth (Fréchet smooth or a continuous Gâteaux smooth) function f from X onto Y such that f vanishes outside a bounded set and all the derivatives of f are surjections. In particular we deduce the following results. For the Gâteaux case, when X and Y are separable and X is infinite-dimensional, there exists a continuous Gâteaux smooth function f from X to Y , with bounded support, so that f (X) = L(X, Y). In the Fréchet case, we get that if a Banach space X has a Fréchet smooth bump and dens X = dens L(X, Y), then there is a Fréchet smooth function f : X −→ Y with bounded support so that f (X) = L(X, Y). Moreover, we see that if X has a C p smooth bump with bounded derivatives and dens X = dens L m s (X; Y) then there exists another C p smooth function f : X −→ Y so that f (k) (X) = L k s (X; Y) for all k = 0, 1, ..., m. As an application, we show that every bounded starlike body on a separable Banach space X with a (Fréchet or Gâteaux) smooth bump can be uniformly approximated by smooth bounded starlike bodies whose cones of tangent hyperplanes fill the dual space X *. In the non-separable case, we prove that X has such property if X has smooth partitions of unity.
A note on the Denjoy-Bourbaki theorem
2003
We prove the following extension of the Mean Value Theorem. Let E be a Banach space and let F : [a, b] → E and ϕ : [a, b] → R be two functions for which there exists a subset A ⊂ [a, b] such that: i) F and ϕ have negligible variation on A,
On Peano's theorem in Banach spaces
2010
We show that if X is an infinite-dimensional separable Banach space (or more generally a Banach space with an infinite-dimensional separable quotient) then there is a continuous mapping f : X → X such that the autonomous differential equation x ′ = f (x) has no solution at any point.
On the range of the derivative of a smooth function and applications
2006
We survey recent results on the structure of the range of the derivative of a smooth real valued function f defined on a real Banach space X and of a smooth mapping F between two real Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of L(X, Y) for the existence of a Fréchet-differentiable mapping F from X into Y so that F (X) = A. Whenever F is only assumed Gâteaux-differentiable, new phenomena appear : we discuss the existence of a mapping F from a Banach space X into a Banach space Y , which is bounded, Lipschitz-continuous, and so that for all x, y ∈ X, if x = y, then F (x) − F (y) L(X,Y) > 1. Applications are given to existence and uniqueness of solutions of Hamilton-Jacobi equations. ESTRUCTURA DEL RANGO DE LA DERIVADA DE UNA FUNCIÓN Resumen. Recogemos recientes resultados sobre la estructura del rango de la derivada de una función real f definida en un espacio de Banach real X y de una aplicación diferenciable F entre dos espacios de Banach reales X e Y. Listamos algunas condiciones necesarias y otras suficientes acerca de un subconjunto A de L(X, Y) para la existencia de una aplicación diferenciable Fréchet F de X en Y de modo que F (X) = A. Cuando se supone tan solo que F es Gâteaux diferenciable, aparecen nuevos fenómenos: discutimos la existencia de una aplicación F de un espacio de Banach X en un espacio de Banach Y acotada, Lipschitz-continua, de tal manera que, para todo x, y ∈ X, si x = y, entonces F (x) − F (y) L(X,Y) > 1. Se dan aplicaciones a la existencia de soluciones de ecuaciones de Hamilton-Jacobi.