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A reflexive Banach space X with a basis (ei) is constructed having the property that every monotone basis is block finitely representable in each block basis of X.

We consider Banach space properties that lie between conditions introduced by Bynum and Landes. These properties depend on the metric behavior of weakly convergent sequences. We also investigate the permanence properties of these conditions.

Two examples of asymptotic ℓ 1 \ell _{1} Banach spaces are given. The first, X u X_{u} , has an unconditional basis and is arbitrarily distortable. The second, X X , does not contain any unconditional basic sequence. Both are spaces of the type of Tsirelson’s.

We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, Π 1 1 \Pi ^1_1 comprehension, is needed to prove such basic facts as the existence of the weak- ∗ * closure of any norm-closed subspace of ℓ 1 = c 0 ∗ \ell _1=c_0^* . This is in contrast to earlier work in which theorems of separable Banach space theory were proved in very weak subsystems of second order arithmetic, subsystems which are conservative over Primitive Recursive Arithmetic for Π 2 0 \Pi ^0_2 sentences. En route to our main results, we prove the Krein-Šmulian theorem in A C A 0 \mathsf {ACA}_0 , and we give a new, elementary proof of a result of McGehee on weak- ∗ * sequential closure ordinals.

Generalized second-order derivatives introduced by Rockafellar in finite-dimensional spaces are extended to convex functions in reflexive Banach spaces. Parallel results are shown in the infinite-dimensional case. A result that plays an important role in applications is that the generalized second-order differentiability is preserved under the integral sign.

Two Banach ∗ ^{\ast } -algebras are naturally associated with a locally compact group, G G : the group algebra, L 1 ( G ) {L^1}(G) , and the measure algebra, M ( G ) M(G) . Either of these Banach algebras is a complete set of invariants for G G . In any Banach ∗ ^{\ast } -algebra, A A , the norm one unitary elements form a group, S S . Using S S we characterize those Banach ∗ ^{\ast } -algebras, A A , which are isometrically ∗ ^{\ast } -isomorphic to M ( G ) M(G) . Our characterization assumes that A A is the dual of some Banach space and that its operations are continuous in the resulting weak ∗ ^{\ast } topology. The other most important condition is that the convex hull of S S must be weak ∗ ^{\ast } dense in the unit ball of A A . We characterize Banach ∗ ^{\ast } -algebras which are isomerically isomorphic to L 1 ( G ) {L^1}(G) for some G G as those algebras, A A , whose double centralizer algebra, D ( A ) D(A) , satisfies our characterization for M ( G ) M(G) . In addition we require A A to consist of those elements of D ( A ) D(A) on which S S (defined relative to D ( A ) D(A) ) acts continuously with its weak ∗ ^{\ast } topology. Using another characterization of L 1 ( G ) {L^1}(G) we explicitly calculate the above isomorphism between A A and L 1 ( G ) {L^1}(G) .

Given Banach spaces E E and F F , a Banach space G E F {G_{EF}} is presented in which E E is embedded and which seems a natural space to which extend F F -valued analytic functions. Any F F -valued analytic function defined on a subset U U of E E may be extended to an open neighborhood of U U in G E F {G_{EF}} . This extension generalizes that of Aron and Berner. It is also related to the Arens product in Banach algebras, to the functional calculus for bounded linear operators, and to an old problem of duality in spaces of analytic functions. A characterization of the Aron-Berner extension is given in terms of continuity properties of first-order differentials.

In this paper, we discuss the possibility of developing a nice i.e. first order theory for Banach spaces: the restrictions on the set of sentences for recent compactness arguments applied to Banach spaces as well as for other model-theoretic results are both natural and necessary; without them we essentially get a second order logic with quantification over countable sets. Especially, the Hanf number for sets of sentences of the first order theory of Banach spaces is exactly the Hanf number for the second order logic of binary relations (with the second order quantifiers ranging over countable sets). In recent years, various authors have tried to develop a first order theory of Banach spaces and have obtained several successful results (2), (3), (6). Nevertheless, they could not find a complete analogy with first order logic. Either, they had to restrict themselves to a proper subset of the set of all formulas of the first order language they considered,1 or else their results applied to classes of normed spaces and did not yield specific information on the Banach spaces included in these classes. It is intuitively clear that the notion of Banach space is not first-order as it involves quantification over sequences. The present paper is an attempt to measure the gap between this notion and what can be expressed by first order logic. Essentially, we show that if one adds to a first order language suitable to discuss normed spaces a single formula meaning "Every Cauchy sequence has a limit", then one gets all the strength of second order logic where all second order variables range over countable sets. Especially, it becomes possible to interpret the notion of well ordering. Let L be a first order language with equality which includes, besides variables, a binary function symbol + and a unary predicate symbol B. Any

We study intersection properties of balls in Banach spaces using a new technique. With this technique we give new and simple proofs of some results of Lindenstrauss and others, characterizing Banach spaces with L 1 ( μ ) {L_1}(\mu ) dual spaces by intersection properties of balls, and we solve some open problems in the isometric theory of Banach spaces. We also give new proofs of some results of Alfsen and Effros characterizing M-ideals by intersection properties of balls, and we improve some of their results. In the last section we apply these results on function algebras, G-spaces and order unit spaces and we give new and simple proofs for some representation theorems for those Banach spaces with L 1 ( μ ) {L_1}(\mu ) dual spaces whose unit ball contains extreme points.

We give an estimate relating the stable type p p constant of a Banach space X X with the dimension of the l p n l_p^n -subspaces of X X . Precisely, let C C be this constant and assume 1 > p > 2 1 > p > 2 . We show that, for each ε > 0 , X \varepsilon > 0,X must contain a subspace ( 1 + ε ) (1 + \varepsilon ) -isomorphic to l p k l_p^k , for every k k less than δ ( ε ) C p ′ \delta (\varepsilon ){C^{p’}} where δ ( ε ) > 0 \delta (\varepsilon ) > 0 is a number depending only on p p and ε \varepsilon .

We show that for a measure γ \gamma on a Banach space directional differentiability implies quasi-translation invariance. This result is shown to imply the Cameron-Martin theorem. A second application is given in which γ \gamma is the image of a Gaussian measure under a suitably regular map.

The present paper is an approach to the local theory of Banach spaces via the ultrapower construction. It includes a detailed study of ultrapowers and their dual spaces as well as a definition of a new notion, the notion of a u-extension of a Banach space. All these tools are used to give a unified definition of many classes of Banach spaces characterized by local properties (such as the L p {\mathcal {L}_p} -spaces). Many examples are given; also, as an application, it is proved that any L p {\mathcal {L}_p} -space, 1 > p > ∞ 1 > p > \infty , has an ultrapower which is isomorphic to an L p {L_p} -space.

A short proof is given of the following known result. For all k, r, t there exists n so that if the t-spaces of an n-space are r-colored there exists a k-space all of whose t-spaces are the same color. Here t-space refers initially to a t-dimensional affine space over a fixed finite field. The result is also shown for a more general notion of t-space.

Let X X and Y Y be quasi complementary subspaces of a separable Banach space B B and let ( z n ) ({z_n}) be a sequence complete in X X . Then (a) there exists a uniformly minimal norming M M -basis ( x n ) ({x_n}) of X X with x m ∈ span ⁡ ( z n ) n ⩾ q m {x_m} \in \operatorname {span} {({z_n})_{n \geqslant {q_m}}} for every m m , q m → ∞ {q_m} \to \infty ; (b) if ( x n ) ({x_n}) is a uniformly minimal norming M M -basis of X X , there exists a uniformly minimal norming M M -basis of B B which is an extension of ( x n ) ({x_n}) ; (c) there exists a uniformly minimal norming M M -basis ( x n ) ∪ ( y n ) ({x_n}) \cup ({y_n}) of B B with ( x n ) ⊂ X ({x_n}) \subset X and ( y n ) ⊂ Y ({y_n}) \subset Y .

Combining combinatorial methods from set theory with the functional structure of certain Banach spaces we get some results on the isomorphic structure of nonseparable Banach spaces. The conclusions of the paper, in conjunction with already known results, give complete answers to problems of the theory of Banach spaces. An interesting point here is that some questions of Banach spaces theory are independent of Z.F.C. So, for example, the answer to a conjecture of Pełczynski that states that the isomorphic embeddability of L 1 { − 1 , 1 } α {L^1}{\{ - 1,\,1\} ^\alpha } into X ∗ {X^{\ast }} implies, for any infinite cardinal α \alpha , the isomorphic embedding of l α 1 l_\alpha ^1 into X X , gets the following form: if α = ω \alpha = \omega , has been proved from Pełczynski; if α > ω + \alpha > {\omega ^ + } , the proof is given in this paper; if α = ω + \alpha = {\omega ^ + } , in Z .F .C . + C .H . {\text {Z}}{\text {.F}}{\text {.C}}{\text {.}} + {\text {C}}{\text {.H}}{\text {.}} , an example discovered by Haydon gives a negative answer; if α = ω + \alpha = {\omega ^ + } , in Z .F .C . + ⌝ C .H . + M .A . {\text {Z}}{\text {.F}}{\text {.C}}{\text {.}} + \urcorner {\text {C}}{\text {.H}}{\text {.}} + {\text {M}}{\text {.A}}{\text {.}} , is also proved in this paper.

Given two Banach spaces F | | F|| and X | | | | X||\,|| , write F fr X iff F{\text { fr }}X{\text { iff}} for each finite-dimensional subspace F ′ F’ of F F and each number ε > 0 \varepsilon > 0 , there is an isomorphism V V of F ′ F’ into X X such that | | x | − | | V x | | | ≤ ε ||x| - ||Vx||| \leq \varepsilon for each x x in the unit ball of F ′ F’ . Given a property P {\mathbf {P}} of Banach spaces, X X is called super- P iff F fr X {\mathbf {P}}{\text { iff }}F{\text { fr }}X implies F F is P {\mathbf {P}} . Ergodicity and stability were defined in our articles On B B -convex Banach spaces, Math. Systems Theory 7 (1974), 294-299, and C. R. Acad. Sci. Paris Ser. A 275 (1972), 993, where it is shown that super-ergodicity and super-stability are equivalent to super-reflexivity introduced by R. C. James [Canad. J. Math. 24 (1972), 896-904]. Q Q -ergodicity is defined, and it is proved that super- Q Q -ergodicity is another property equivalent with super-reflexivity. A new proof is given of the theorem that J J -spaces are reflexive [Schaffer-Sundaresan, Math. Ann. 184 (1970), 163-168]. It is shown that if a Banach space X X is B B -convex, then each bounded sequence in X X contains a subsequence ( y n ) ({y_n}) such that the Cesàro averages of ( − 1 ) i y i {( - 1)^i}{y_i} converge to zero.

The present paper is devoted to a representation theorem of central double centralizers on a complex Banach algebra with a bounded approximate identity. In particular, our result implies the representation theorem of the ideal center of an arbitrary C ∗ {C^\ast } -algebra established by J. Dixmier.

We consider the problem of decomposing a Banach lattice Z as a direct sum Z = X @ Y where X and Y are complemented subspaces satisfying a condition of incomparability (e.g. every operator from Y to X is strictly singular). We treat both the atomic and nonatomic cases. In particular we answer a question of Wojtaszczyk by showing that L1 fflL2 has unique structure as a nonatomic Banach lattice. One of the most important problems in the theory of Banach lattices, which is still open, is whether any complemented subspace of a Banach lattice must be linearly isomorphic to a Banach lattice. The main difficulty seems to lie in the fact that most of the criteria for a Banach space to be isomorphic to a lattice do not really distinguish between lattices and their complemented subspaces. We do not actually treat this question in the present paper but rather consider the situation Z = X d3 Y, where Z is a Banach lattice and X and Y two complemented subspaces which are assumed to satisfy different conditions that make them "distinct" in some or another sense. This line of research was initiated by P. Wojtaszczyk 128] (and also by I. S. Edelstein and P. Wojtaszczyk [3]) who proved that if Z has a normalized unconditional basis {Zn}n=l (i.e. it is a separable atomic lattice) so that every linear operator from Y into X is compact then {zn}n=l splits into two disjoint parts which are respectively equivalent to bases of X and Y. In particular, both X and Y have unconditional bases. The proof of this result is based on a fundamental theorem from [28 and 3], which is mentioned below as Theorem A. We give here a different proof which does not make use of Theorem A but instead is based on a simple "change of signs" result from [2], which is described below as Theorem B. We also consider the case when the compactness assumption above is replaced by the total incomparability of X and Y for which we prove a similar result provided X and Y have unconditional bases. Unfortunately, the most interesting case when every operator from Y into X is assumed to be strictly singular (which was raised as an open problem in [28]) remains unsolved. We conclude the section devoted to the atomic case with a simple theorem on block bases of a space with unconditional basis {zn}°°=1 whose span is complemented. Such a block basis splits into two disjoint parts, the first equivalent to a subsequence of {zn}n= Received by the editors September 5, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46B30, 46B15; Secondary 46A40. This research was supported by Grant No. 84-00210 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. The research of the first author was also supported in part by NSF Grant DMS 8500938. The research of the second author was also supported in part by NSF Grant DMS 8601401. (r) 1987 American Mathematical Society OOO2-9947/87 $1.00 + $.25 per page

Let E E be a Banach space. One often wants to measure how far E E is from being a Hilbert space. In this paper we define the skewness s ( E ) s(E) of a Banach space E E , 0 ⩽ s ( E ) ⩽ 2 0 \leqslant s(E) \leqslant 2 , which describes the asymmetry of the norm. We show that s ( E ) = s ( E ∗ ) s(E) = s({E^{\ast }}) for all Banach spaces E E . Further, s ( E ) = 0 s(E) = 0 if and only if E E is a (real) Hilbert space and s ( E ) = 2 s(E) = 2 if and only if E E is quadrate, so s ( E ) > 2 s(E) > 2 implies E E is reflexive. We discuss the computation of s ( L p ) s({L^p}) and describe its asymptotic behavior near p = 1 , 2 p = 1,2 and ∞ \infty . Finally, we discuss a higher-dimensional generalization of skewness which gives a characterization of smooth Banach spaces.

Let ∑ n ∈ ω h ( n ) \sum \nolimits _{n\,\, \in \,\omega } {h(n)} be a conditionally convergent series in a real Banach space B. Let S ( h ) S(h) denote the set of sums of the convergent rearrangements of this series. A well-known theorem of Riemann states that S ( h ) = B S(h)\, = \,B if B = R B\, = \,R , the reals. A generalization of Riemann’s Theorem, due independently to Levy [L] and Steinitz [S], states that if B is finite dimensional, then S ( h ) S(h) is a linear manifold in B of dimension > 0 > \,0 . Another generalization of Riemann’s Theorem [M] can be stated as an instance of the Levy-Steinitz Theorem in the Banach space of regulated real functions on the unit interval I. This instance generalizes to the Banach space of regulated B-valued functions on I, where B is finite dimensional, implying a generalization of the Levy-Steinitz Theorem.

ABSTRACr. Existence theorems are obtained for optimization problems where the cost functional takes values in an ordered Banach space. The order is defined in terms of a closed convex cone in the Banach space; and in this connection, several relevant properties of cones are studied and they are shown to coincide in the finite dimensional case. The notion of a weak (Pareto) extremum of a subset of an ordered Banach space is then introduced. Existence theorems are proved for extrema for Mayer type as well as Lagrange type problems-in a manner analogous to and including those with scalar valued cost. The side conditions are in the form of general operator equations on a class of measurable functions defined on a finite measure space. Needed closure and lower closure theorems are proved. Also, several analytic criteria for lower closure are provided. Before the appendix, several illustrative examples are given. In the appendix, a criterion (different from the one used in main text) is given and proved, for the Pareto optimality of an element.

The theory of complex interpolation of Banach spaces is viewed as a branch of the theory of vector valued holomorphic functions. Versions of the Schwarz lemma, Liouville’s theorem, the identity theorem and the reflection principle are proved and are interpreted from the point of view of interpolation theory.

It is proved that every Banach space with a separable dual embeds into a space with a shrinking basis. It follows that every separable reflexive space can be embedded in a reflexive space with a basis.

Multivariate nonincreasing rearrangement and averaging functions are defined for functions defined over product spaces. An investigation is made of Banach function spaces with mixed norms and using multivariate rearrangements. Particular emphasis is given to the L ( P , Q ; ∗ ) L(P,Q;\ast ) spaces. These are Banach function spaces which are in terms of mixed norms, multivariate rearrangements and the Lorentz L ( p , g ) L(p,g) spaces. Embedding theorems are given for the various function spaces. Several well-known theorems are extended to the L ( P , Q ; ∗ ) L(P,Q;\ast ) spaces. Principal among these are the Strong Type (Riesz-Thorin) Interpolation Theorem and the Convolution (Young’s inequality) Theorem.

Theorems of the Bochner-Sazonov type are proved for Banach spaces with a basis. These theorems give sufficient conditions of a topological nature under which a positive definite function is the characteristic functional of a probability measure. The conditions are, in a certain natural sense, best possible. Central limit theorems of the Lindeberg type for triangular systems of random variables taking values in a Banach space with a basis are obtained. Applications to I, and C[0, 1] are given. 0. Introduction. In this work we will be concerned with the following problems which we describe in general terms. (1) Generalization of Bochner's theorem. Let X be a topological vector space X' its dual space. Let g: X' -C be positive definite, g(O)= 1. Under what conditions, or more specifically, under what topological conditions is g the characteristic functional of a probability measure on X? (2) Generalization of the central limit theorem. Let {Zj}, j= 1, 2, . . ., be an independent sequence of X-valued random variables with a common distribution. Suppose for each y E X', E =0 and E 2 < oo. Under what conditions (on the common distribution) does Y(n 1/2 1 Zj) converge weakly? More generally, when are limit theorems of the Lindeberg type valid? For the first problem, the classical theorem of Bochner provides a complete answer when the space X is finite dimensional: g is a characteristic function if and only if g is continuous at 0. (See, for example, [11, p. 207]). A complete answer is also possible when the space is a locally compact abelian group. (See, for instance, [17].) In the infinite-dimensional case, Kolmogorov showed that g is the FourierStieltjes transform of afinitely additive measure defined on the field of cylinder sets induced by X' and u-additive on each u-field induced by a finite subset of X' if and only if the restriction of g to each finite-dimensional subspace of X' is Received by the editors October 10, 1969. AMS subject classifications. Primary 6008, 2846; Secondary 6030.

Let A A be a complex ∗ ^ \ast -algebra. If f f is a positive functional on A A , let I f = { x ∈ A : f ( x ∗ x ) = 0 } {I_f} = \{ x \in A:f(x^ \ast x) = 0\} be the corresponding left ideal of A A . Set P = ∩ I f P = \cap {I_f} , where the intersection is over all positive functionals on A A . Then A A is called P P -commutative if x y − y x ∈ P xy - yx \in P for all x , y ∈ A x,y \in A . Every commutative ∗ ^ \ast -algebra is P P -commutative and examples are given of noncommutative ∗ ^ \ast -algebras which are P P -commutative. Many results are obtained for P P -commutative Banach ∗ ^ \ast -algebras which extend results known for commutative Banach ∗ ^ \ast -algebras. Among them are the following: If A 2 = A {A^2} = A , then every positive functional on A A is continuous. If A A has an approximate identity, then a nonzero positive functional on A A is a pure state if and only if it is multiplicative. If A A is symmetric, then the spectral radius in A A is a continuous algebra seminorm.

The direct product Ei®nE2 of two Banach spaces Eu E2 has been defined before [5](2) as the closure of the normed linear set $In(Ei, E2) (that is, linear set 3i(£i, £2) of expressions 22Â-ifi®4>i, hi which N is a norm) [5, p. 200, Definition 1.3] and [6, p. 499, b]. Let N denote a crossnorm whose associate N' is also a crossnorm [5, p. 208]. Then, the cross-space Ei®nEi determines uniquely a "conjugate space" (Ei®NE2)' and an "associate space" Ei ®n'E2 . It is shown [5, p. 205] that Ei ®N'E2 is always included in (Ei®nE2)'. While there are many known examples of cross-spaces for which the associate space coincides with the conjugate space—for example, the cross-space generated by the self-associate crossnorm constructed for Hubert spaces by F. J. Murray and John von Neumann [3, p. 128] and [5, pp. 212-214]—it is not without interest to construct a cross-space for which the associate space forms a proper subset of the conjugate space (§§1-2). For reflexive Banach spaces Ei, E2 (that is, such that 25/' =Ei), and a reflexive crossnorm N [6, p. 500], the reflexivity of Ei®^E2 implies (Ei®nE2)'—Ei ®N'Ei [6, p. 505]. Thus, the finding of the exact conditions imposed upon reflexive Banach spaces and a reflexive crossnorm for which the resulting cross-space is reflexive is closely connected with the above-mentioned problem. In §1, we show that for a "natural crossnorm" N, L'®nL' is a proper subset of (L®NL)'. In §2 we prove that for a "natural crossnorm" N, V®ud' is a proper subset of (/ ® nI) '• In §3 we show that for any p > 1, lp ® nIi is not reflexive, provided l/p + l/q = 1 and N denotes the least crossnorm whose associate is also a crossnorm [5, p. 208]. The last one is reflexive [6, p. 501 ]. 1. Let Z,(i) and L$) denote the Banach spaces of all functions integrable in the sense of Lebesgue on the interval Ogsgl, and on the square 0^s, t ^ 1 respectively. Similarly, let M(u and M denote the Banach spaces of all functions Lebesgue measurable and essentially bounded on the interval OiSs^l and the square 0^s, f gl respectively [l, pp. 10, 12]. We recall that

A Banach space is flat if the girth of its unit ball is 4 and if the girth is achieved by some curve. (Equivalently, its unit ball can be circumnavigated along a centrally symmetric path whose length is 4.) Some basic geometric properties of flat Banach spaces are given. In particular, the term flat is justified.

Introduction. In a previous paper [7](1), for two Banach spaces E1, E2, the Banach spaces E1?E2, E; X E2, E" G E2 ' [7, p. 205 ] are constructed. If the norm N [7, Definition 3.1] is defined on E1?E2, then the associate norm N' [7, Definition 3.2 and Lemma 3.11 is defined on E; X E2. Similarly N" denotes the norm on EP X E2'. Among the unsolved problems (mentioned in [7, ?6]), are listed the following two: A. What are the exact conditions imposed upon a crossnorm [7, Definition 3.3] under which (E10 E2)'=E; XE2' holds? B. rs the associate with every crossnorm also a crossnorm, or do there exist crossnorms whose associates are not crossnorms? In the present paper we present a "partial" answer to problem A (which we denote by A*), and a "partial" answer to problem B (which we denote by B*). A*. A uniformly convex crossnorm N sets up the relation (E1l0E2)' E' 0EX if, and only if, N"= N. B*. For reflexive Banach spaces (that is, such that EP = E1, E' =E2) the associate with every crossnorm is also a crossnorm. In this paper we also show that the values of a crossnorm for all expressions of rank not greater than 2 do not necessarily determine the crossnorm. The following should be mentioned in immediate connection with problem A: It is evident that for norms for which (E1l E2)'= E( XE' holds, N" = N. Since in general (for any norm N) all we can state is (E1l?E2)'DE; ?El [7, p. 205], we have no basis for assuming that N" represents the norm in (E1l E2)", or N" =N for expressions in (E1, E2)C2[(E', E2') [7, Definition 1.3]. Therefore, N"

For a contingent differential equation that takes values in the closed, convex, nonempty subsets of a Banach space E, we prove an existence theorem and we investigate the extendability of solutions and the closedness and continuity properties of solution funnels. We consider first a space E that is separable and reflexive and then a space E with a separable second dual space. We also consider the special case of a pointvalued or ordinary differential equation. 0. Introduction. Consider the contingent differential equation

Let L : D ⊂ X → D ⊂ X ∗ L:D \subset X \to D \subset {X^ \ast } be a densely defined linear map of a reflexive Banach space X to its conjugate X ∗ {X^\ast } . Define M and M ∗ {M^\ast } to be the respective null spaces of L and its formal adjoint L ∗ {L^\ast } . Let f : X → X ∗ f:X \to {X^\ast } be continuous. Under certain conditions on L ∗ {L^\ast } and f there exist weak solutions to L u = f ( u ) Lu = f(u) provided for each w ∈ X , v ( w ) ∈ M w \in X,v(w) \in M can be found such that f ( v ( w ) + w ) f(v(w) + w) annihilates M ∗ {M^ \ast } . Neither M and M ∗ {M^\ast } nor their annihilators need be the ranges of continuous linear projections. The results have applications to periodic solutions of partial differential equations.

Introduction. The present paper is concerned with the general problem of extending the classical theory of analytic functions of a complex variable. This question received the attention of Hilbert and F. Riesz, and probably goes back to Volterra. More recently N. Dunford, L. Fantappie, I. Gelfand, E. R. Lorch, A. D. Michal, and A. E. Taylor have contributed to the subject (see bibliography). Our approach differs from most of the others in two main respects, namely, in the type of domain and range of the functions and in the definition of analyticity. We consider functions which have for their domains and ranges subsets of an abstract commutative Banach algebra with unit and we use a definition of analyticity introduced by E. R. Lorch [1]. It is known [4] that a function analytic by this definition is differentiable in the Frechet sense but not every Frechet-differentiable function on a commutative Banach algebra is analytic in the Lorch sense. Accordingly, the Lorch theory is the richer. For the most part, the development of the primary aspects of the Lorch theory parallels that of the classical theory. Interesting departures occur in the more advanced stages. As one would expect, the Cauchy integral theorem and formula occupy a central position and yield the Taylor expansion in the usual way. With Lorch's work as a foundation, we have extended the theory to include a study of Laurent expansions and analytic continuation. There are also some results on the zeros of polynomials over the algebra, on rational functions and their integrals, and on the singularities of analytic functions. Although the objective of this investigation was essentially analytical, we have also obtained results of an algebraic-topological character (e.g. distribution of singular elements). This was a natural outcome of the algebraic character of the techniques used: 1. Basic concepts. A set, B, of elements (denoted by Latin letters a, b, c, x, y, z, ) is a "Banach algebra" if (1) B is an algebra over the complex numbers (denQted by greek letters), (2) B is a complex Banach space, and (3) the norm satisfies the inequality Ilabil

which we shall call local uniform convexity. Geometrically this differs from uniform convexity in that it is required that one end point of the variable chord remain fixed. In section I we prove a general theorem on the product of locally uniformly convex Banach spaces and with the aid of this theorem we establish that the two notions are actually different. Section II is devoted to the investigation of the relationship between local uniform convexity and strong differentiability of the norm. In section III we investigate conditions for isomorphism of a Banach space with a locally uniformly convex space.

John R. Rice [Approximation of functions. Vol. II, Addison-Wesley, New York, 1969] investigated best approximation from a nonlinear manifold in a finite dimensional, smooth, and rotund space. The authors define the curvature of a manifold by comparing the manifold with the unit ball of the space and suitably define the “folding” of a manifold. Rice’s Theorem 11 extends as follows: Theorem. Let X be a uniformly smooth Banach space, and F : R n → X F:{R^n} \to X be a homeomorphism onto M = F ( R n ) M = F({R^n}) . Suppose ∇ F ( a ) \nabla F(a) exists for each a in X, ∇ F \nabla F is continuous as a function of a, and ∇ F ( a ) ⋅ R n \nabla F(a) \cdot {R^n} has dimension n. Then, if M has bounded curvature, there exists a neighborhood of M each point of which has a unique best approximation from M. A variation theorem was found and used which locates a critical point of a differentiable functional defined on a uniformly rotund space Y. [See M. S. Berger and M. S. Berger, Perspectives in nonlinearity, Benjamin, New York, 1968, p. 58ff. for a similar result when Y = R n Y = {R^n} .] The paper is concluded with a few remarks on Chebyshev sets.

0. Introduction. The purpose of this paper is to study the spectral theory of a closed linear transformation T on a reflexive Banach space B. This will be done by means of certain vector-valued measures which are related to the transformation. (A set function m from the Borel sets of the complex plane to B will be called a vector-valued measure if the series EJ= j m (Si) converges to mr(Ui Si) for every sequence { Si I of disjoint Borel sets. The relevant properties of vector-valued measures are briefly derived in ?1(2). A vector-valued measure m will be called a T-measure if Tm(S) =fszdm(z) for all bounded Borel sets S. The properties of T-measures are studied in ?2. The results of ?2 are applied in ?3 to a class of transformations which have been called scalar-type transformations by Dunford [5], and which we call simply scalar transformations. A scalar operator as defined by Dunford is essentially one which admits a representation of the type t = fzdE(z), where E is a spectral measure. Unbounded scalar transformations have been studied by Taylor [16]. The main result of ?3 is Theorem 3.2, in which properties of the closures of certain sets of scalar transformations are derived. This theorem is actually a rather general spectral-type theorem, which has applications to several problems in the theory of linear transformations. As a corollary we obtain a well-known theorem, which might be called the spectral theorem for symmetric transformations, as given in Stone [15]. We also derive as a corollary the spectral theorem for self-adjoint transformations. Other corollaries of Theorem 3.2, which apply to results of Bade [2] and Halmos [9] are derived. In ?4 a functional calculus is developed for a class r of transformations T for which both T-measures and T*-measures exist in sufficient abundance. This is a very general functional calculus, so that correspondingly the usual theorems of functional calculus must be weakened if they are to remain true. There is a generalization of the concept of a T-measure introduced in ?5. Many theorems proved in ?2 have analogues which hold after the generalization. The new type of vector-valued measures have much the same relation

This paper is concerned with the quasi-linear evolution equation u'(t) + A(t, u(t))u(t) = 0 in [0, T], u(0) = xo in a Banach space setting. The spirit of this inquiry follows that of T. Kato and his fundamental results concerning linear evolution equations. We assume that we have a family of semigroup generators that satisfies continuity and stability conditions. A family of approximate solutions to the quasi-linear problem is constructed that converges to a "limit solution." The limit solution must be the strong solution if one exists. It is enough that a related linear problem has a solution in order that the limit solution be the unique solution of the quasi-linear problem. We show that the limit solution depends on the initial value in a strong way. An application and the existence aspect are also addressed. This paper is concerned with the quasi-linear evolution equation u'(t) + A(t, u(t))u(t) = 0 in[O, T], u(O) = xo in a Banach space setting. The spirit of this inquiry follows that of T. Kato. Kato wrote a fundamental paper on linear evolution equations in 1953 [9]; that is, investigation of u'(t) + A(t)u(t) = 0 on[O, T], u(0) = x0. He strengthened and extended his analysis of the linear problem in 1970 [11]. Kato also wrote on the quasi-linear problem in 1975 [13]. We feel that our results give a natural approach to dealing with the quasi-linear problem. After discussing the setting and method of attack, our theorem is stated and proved. We then give an application of the theorem using the Sobolevskii-Tanabe theory of linear evolution equations of parabolic type. A proposition relevant to our theorem is also given. Let X and Y be Banach spaces, with Y densely and continuously embedded in X. Let x0 E Y, T > 0, r > r, > 0, r2 > 0, W= Bx(xo; r), Z = Bx(xo; rl) n By(xo; r2), and for each t E [0, T] and w E W, let -A (t, w) be the infinitesimal generator of a strongly continuous semigroup of bounded linear operators in X, with Y c D(A(t, w)). We consider the quasi-linear evolution equation v'(t) + A(t, v(t))v(t) = 0. (QL) Received by the editors December 14, 1978 and, in revised form, July 6, 1979. AMS (MOS) subject classifications (1970). Primary 34G05, 47D05; Secondary 65J05, 41A65.

We define a function F on a Banach space V to be locally ε \varepsilon -supported by u ∗ ∈ V ∗ {u^\ast } \in {V^\ast } at u ∈ V u \in V if there exists an η > 0 \eta > 0 such that ‖ v − u ‖ ⩽ η ⇒ F ( v ) ⩾ F ( u ) + ⟨ u ∗ , v − u ⟩ − ε ‖ v − u ‖ \left \| {v - u} \right \| \leqslant \eta \Rightarrow F(v) \geqslant F(u) + \langle {u^\ast },v - u\rangle - \varepsilon \left \| {v - u} \right \| . We prove that if the Banach space V admits a nonnegative Fréchet-differentiable function with bounded nonempty support, then, for any > 0 > 0 and every lower semicontinuous function F, there is a dense set of points u ∈ V u \in V at which F is locally ε \varepsilon -supported. The applications are twofold. First, to the study of functions defined as pointwise infima; we prove for instance that every concave continuous function defined on a Banach space with Fréchet-differentiable norm is Fréchet-differentiable generically (i.e. on a countable intersection of open dense subsets). Then, to the study of optimization problems depending on a parameter u ∈ V u \in V ; we give general conditions, mainly in the framework of uniformly convex Banach spaces with uniformly convex dual, under which such problems generically have a single optimal solution, depending continuously on the parameter and satisfying a first-order necessary condition.

We characterize conjugate Banach spaces X* having the Radon-Nikodym Property as those spaces such that any separable subspace of X has a separable conjugate. Several applications are given. Introduction. There are several equivalent formulations of the Radon-Nikodym Property (RNP) in Banach spaces; we give perhaps the earliest definition: a Banach space X has RNP if given any finite measure space (S, E, p) and any X valued measure m on E, with m having finite total variation and being absolutely continuous with respect to ,, then m is the indefinite integral with respect to ,u of an X valued Bochner integrable function on S. The first study of this property was by Dunford and Pettis [4] and Phillips [11] (see also [5]). It follows from the work of Dunford and Pettis and Phillips that reflexive Banach spaces and separable conjugate spaces have RNP. More generally, the following is true: THEOREM A. If X is a Banach space such that for any separable subspace Y of X, Y* is separable, then X* has RNP. The above result was observed by Ubl [15] and also can be obtained from a result of Grothendieck (Theorem B below). The first characterizations of RNP were given by Grothendieck in [6]. Grothendieck's approach, the one we shall use, is that of studying certain classes of operators. An operator T: X -+ Y is a continuous linear function T from the Banach space X to the Banach space Y. An operator T: X -> Y is said to be an integral operator if there exist a compact Hausdorff space K, a Radon measure ,u on K, and operators R, and S, such that Received by the editors July 24, 1973 and, in revised form, February 2, 1974. AMS (MOS) subject classifications (1970). Primary 28A45, 46G10, 46B99.

The problem of existence of solutions to the initial value problem x' = f(t, x), x(to) = x0 E F, where f E C[[t0, t0 + al X F, El, F is a locally closed subset of a Banach space E is considered. Nonlinear comparison functions and dissipative type conditions in terms of Lyapunov-like functions are employed. A new comparison theorem is established which helps in surmounting the difficulties that arise in this general setup.