Degree theory forC 1 Fredholm mappings of index 0

We develop a degree theory forC ¹ Fredholm mappings of index 0 between Banach spaces and Banach manifolds. As in earlier work devoted to theC ² case, our approach is based upon the concept of parity of a curve of linear Fredholm operators of index 0. This avoids considerations about Fredholm structures involved in other approaches and leads to a theory as complete as that of Leray-Schauder in a much broader setting. In particular, the well-known possible sign change under homotopy is fully elucidated. The technical difficulty arising withC ¹ versusC ² Fredholm mappings of index 0 is notorious: with onlyC ¹ smoothness, the Sard-Smale theorem is no longer available to handle crucial issues involving homotopy. In this work, this difficulty is overcome by using a new approximation theorem forC ¹ Fredholm mappings of arbitrary index instead of the Sard—Smale theorem when dealing with homotopies.     

Local conjugacy theorem, rank theorems in advanced calculus and a generalized principle for constructing Banach manifolds

Applications of locally fine property for operators are further developed. LetE andF be Banach spaces andF:U(x ₀)⊂E→F be C¹ nonlinear map, whereU (x ₀) is an open set containing pointx ₀∈E. With the locally fine property for Frechet derivativesf′(x) and generalized rank theorem forf′(x), a local conjugacy theorem, i. e. a characteristic condition forf being conjugate tof′(x ₀) near x₀,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced calculus are established, including a theorem for C¹ Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives rise to a generalized principle for constructing Banach submanifolds.     

Sard’s theorem for mappings between Fréchet manifolds

We prove an infinite-dimensional version of Sard’s theorem for Fréchet manifolds. Let M (respectively, N) be a bounded Fréchet manifold with compatible metric d _M (respectively, d _N ) modeled on Fréchet spaces E (respectively, F) with standard metrics. Let f : M → N be an MC ^k -Lipschitz–Fredholm map with k > max{Ind f, 0}: Then the set of regular values of f is residual in N.     

On applications of the Taylor formula in some quasispaces

We consider some metric spaces with quasimetric (quasispaces) comprising uniformly regular (equiregular) Carnot — Carathéodory quasispaces whose quasimetric is induced by C ^ϒ−1-smooth vector fields of formal degree not higher than ϒ. For these spaces, some analogues of the Campbell — Hausdorff formula are derived, which allows us to prove a theorem on a nilpotent tangent cone, a theorem on isomorphism of various nilpotent tangent cones defined at a common point, and a local approximation theorem.     

Uniqueness of <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroups on a general locally convex vector space and an application

The main purpose is to generalize a theorem of Arendt about uniqueness of C ₀-semigroups from Banach space setting to general locally convex vector spaces. More precisely, we show that cores are the only domains of uniqueness for C ₀-semigroups on locally convex spaces. As an application, we find a necessary and sufficient condition for that the mass transport equation has one unique L ¹(ℝ ^d ,dx) weak solution.     

A characterization of constant-time linear control systems satisfying the Pontryagin maximum principle

Using the subdifferential, we extend the main characterization of Banach linear systems satisfying the Pontryagin maximum principle, given in our previous paper (Ref. 1), to the case whenF andX are locally convex spaces and the norm ofF is replaced by an arbitrary continuous convex functionalh onF.     

Moduli of non-dentability and the radon-nikodým property in banach spaces

We show that for a separable Banach spaceX failing the Radon-Nikodym property (RNP), andε > 0, there is a symmetric closed convex subsetC of the unit ball ofX such that every extreme point of the weak-star closure ofC in the bidualX** has distance fromX bigger than 1 −ε. An example is given showing that the full strength of this theorem does not carry over to the non-separable case. However, admitting a renorming, we get an analogous result for this theorem in the non-separable case too. We also show that in a Banach space failing RNP there is, forε > 0, a convex setC of diameter equal to 1 such that each slice ofC has diameter bigger than 1 −ε. Some more related results about the geometry of Banach spaces failing RNP are given.     

An extension theorem for separable Banach spaces

We construct a totally disconnected ω^*, norming subsetF of the unit ballB ^* of an arbitrary separable Banach space,X, and an operator fromC(F) toC(B^*) that “amost” commutes with the natural embeddings ofX. This is used to give a new proof of Milutin's theorem and to prove some new results on complemented subspaces ofC[0, 1] with separable dual. In particular we show that a complemented subspace ofC(ω^ω), is either isomorphic toC(ω^ω) or toc _u.     

Characterizations of some classes ofL 1-preduals by the Alfsen-Effros structure topology

Using the Alfsen-Effros structure topology on the extreme boundary of the dual unit ball of a complex Banach space, we give characterizations ofL ¹-preduals (i.e., Banach spaces whose duals are isometrically isomorphic toL ¹ (μ) for a non-negative measure μ) and some of its subclasses viz.G-spaces,C _σ-spaces and _c ⁰(Г) spaces.     

Structure of spaces ofC ∞-functions on nuclear spaces

LetE be a real nuclear locally convex space; we prove that the space ℰ_ub(E), of allC ^∞-functions of uniform bounded type onE, coincides with the inductive limit of the spaces ℰ_Nbc(E _v) (introduced by Nachbin-Dineen), whenV ranges over a basis of convex balanced 0-neighbourhoods inE. LetE be a real nuclear bornological vector space; we prove that the space ℰ(E) of allC ^∞-functions onE coincides with the projective limit of the spaces ℰ_Nbc(E _B), whenB is a closed convex balanced bounded subset ofE. As a consequence we obtain some density results and a version of the Paley-Wiener-Schwartz theorem. Research done during the stay of this author at the University of Bordeaux (France) in the academic year 1980–1981.     

On convergence of sections of sequences in Banach spaces

An elementary proof of the (known) fact that each element of the Banach spaceℓ _w ^p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ _w ^p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.     

An Isomorphic Version of Nakano’s Characterisation of <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>(Σ)

In 1941 Nakano gave a characterisation of real spaces C₀(Σ) as M-spaces satisfying an additional restriction on their norm. We give an isomorphic version of Nakano’s result which involves showing that for any Banach lattice (a suitably modified version of) Nakano’s condition is equivalent to the norm on the Dedekind completion being Fatou.     

Density and unique decomposition theorems for the lattice of cellular classes

A classC of pointed spaces is called a cellular class if it is closed under weak equivalences, arbitrary wedges and pointed homotopy pushouts. The smallest cellular class containingX is denoted byC(X), and a partial order relation ≪ is defined by:X ≪Y ifY εC(X). In this text we investigate the sub partial order sets generated respectively by simply connected finite CW-complexes and by rational spaces. For rational spaces we prove a unique decomposition theorem, a density theorem and the existence of infinitely many non-comparable elements. We then prove the density theorem for a generic class of finite CW-complexes.     

T 1 theorem for Besov spaces on nonhomogeneous spaces

Suppose μ is a Radon measure on ℝ ^d , which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀>0 such that for all x∈supp(μ) and r>0, μ(B(x, r))⪯C₀rⁿ, where 0<n⪯d. We prove T1 theorem for non doubling measures with weak kernel conditions. Our approach yields new results for kernels satisfying weakened regularity conditions, while recovering previously known Tolsa’s results. We also prove T1 theorem for Besov spaces on nonhomogeneous spaces with weak kernel conditions given in [7].     

Martingales with values in uniformly convex spaces

Using the techniques of martingale inequalities in the case of Banach space valued martingales, we give a new proof of a theorem of Enflo: every super-reflexive space admits an equivalent uniformly convex norm. Letr be a number in ]2, ∞[; we prove moreover that if a Banach spaceX is uniformly convex (resp. ifδ _x(?)/? ^r when? → 0) thenX admits for someq<∞ (resp. for someq<r) an equivalent norm for which the corresponding modulus of convexity satisfiesδ(?)/? ^q → ∞ when? → 0. These results have dual analogues concerning the modulus of smoothness. Our method is to study some inequalities for martingales with values in super-reflexive or uniformly convex spaces which are characteristic of the geometry of these spaces up to isomorphism.     

The classification of injective Banach lattices

LetE be a 1-injective Banach lattice,X any Banach space andT: E ← X a norm bounded linear operator. Then eitherT is an isomorphism on some copy ofl _∞ inE or for all σ > 0 there is φ ∈E ₊ ^′ such that ‖Tu‖≦φ (|u|)+σ ‖u‖ for allu ∈E. We deduce the theorem that: A norm order continuous injective Banach lattice is order isomorphic to an (AL)-space.     

Sur un théorème de J. L. Krivine concernant la caracterisation des classes d’espaces isomorphes a des espaces d’orlicz généralisés et des classes voisines

The classL ^p the classes λ-SL ^p of spaces λ isomorphic to subspaces ofL ^p spaces and λ-QL ^p of subquotients have been characterized in the literature by formulas of certain simple forms. A theorem of Krivine gives a general demonstration of these results in the framework of ψ normed spaces. In particular, characterizations of subspaces and subquotients of certain classes of generalized Orlicz spaces are obtained.      

Complete rank theorem of advanced calculus and singularities of bounded linear operators

Let E and F be Banach spaces, f: U ⊂ E → F be a map of C ^r (r ⩾ 1), x ₀ ∈ U, and ft (x ₀) denote the FréLechet differential of f at x ₀. Suppose that f′(x ₀) is double split, Rank(f′(x ₀)) = ∞, dimN(f′(_{x 0})) > 0 and codimR(f′(x ₀)) s> 0. The rank theorem in advanced calculus asks to answer what properties of f ensure that f(x) is conjugate to f′(x ₀) near x ₀. We have proved that the conclusion of the theorem is equivalent to one kind of singularities for bounded linear operators, i.e., x ₀ is a locally fine point for f′(x) or generalized regular point of f(x); so, a complete rank theorem in advanced calculus is established, i.e., a sufficient and necessary condition such that the conclusion of the theorem to be held is given.      

Weakly stable Banach spaces

The class of stable Banach spaces, inspired by the stability theory in mathematical logic, was introduced by Krivine and Maurey and provided the proper context for the abstract formulation of Aldous’ result of subspaces ofL ¹. In this paper we study the wider class of weakly stable Banach spaces, where the exchangeability of the iterated limits occurs only for sequences belonging to weakly compact subsets, introduced independently by Garling (in an earlier unpublished version of his expository paper on stable Banach spaces brought recently to our attention) and by the authors. Taking into account Rosenthal’s application of the study of pointwise compact sets of Baire-1 functions (Rosenthal compact spaces) in the study of Banach spaces (for whichl ¹ does not embed isomorphically) and of the study of Rosenthal compact sets by Rosenthal and Bourgain-Fremlin-Talagrand, we prove the following analogue of the Krivine-Maurey theorem for weakly stable spaces:If X is infinite dimensional and weakly stable then either l ^p for some p≧1or c_o embeds isomorphically in X (§1). Garling (in the above reference) proved this result under the additional assumption thatX* is separable. We also construct an example of a Banach spaceX which is weakly stable, without an equivalent stable norm, and such thatl ² embeds isomorphically in every infinite dimensional subspace ofX (§3).     

Properties of <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth mappings with one-dimensional gradient range

We find necessary and sufficient conditions for a curve in ℝ ^m×n to be the gradient range of a C ¹-smooth function υ: Ω ⊂ ℝ ⁿ → ℝ ^m . We show that this curve has tangents in a weak sense; these tangents are rank 1 matrices and their directions constitute a function of bounded variation. We prove also that in this case v satisfies an analog of Sard’s theorem, while the level sets of the gradient mapping ▿υ: Ω → ℝ ^m×n are hyperplanes.