the convergence of spline collocation for strongly elliptic equations on curves most boundary element methods for two dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order the equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines the crucial assumption for the generalized boundary integral and integro differential operators is strong ellipticity our analysis is based on simple fourier expansions in particular we extend results by j saranen and w l wendland from constant to variable coefficient equations our results include the first convergence proof of midpoint collocation with piecewise constant functions i e the panel method for solving systems of cauchy singular integral equations

extension theorems of continuous random linear operators on random domains abstract the central purpose of this paper is to prove the following theorem let u be a complete probability space b a normed linear space over the scalar field k e 2 b a separable random domain with linear subspace values and gr e k a continuous random linear operator where gr e x b x e denotes the graph of e then there exists a continuous random linear operator b k such that x x x e and sup x x b x 1 sup x x e x 1 for each in for the case where e is not separable a result similar to the above stated theorem is also given which generalizes and improves many previous results on random generalizations of the hahn banach theorem

random strict convexity and random uniform convexity in random normed modules abstract based on the analysis of stratification structure on random normed modules we first present random strict convexity and random uniform convexity in random normed modules then we establish their respective relations to classical strict and uniform convexity in the process some known important results concerning strict convexity and uniform convexity of lebesgue bochner function spaces can be obtained as a special case of our results further we also give their important applications to the theory of random conjugate spaces as well as best approximation finally we conclude this paper with some remarks showing that the study of geometry of random normed modules will also motivate the further study of geometry of probabilistic normed spaces

boundary element collocation methods using splines with multiple knots we extend the theory of boundary element collocation methods by allowing reduced inter element smoothness or in other words by allowing trial functions that are splines with multiple knots our convergence analysis is based on a recurrence relation for the fourier coefficients of the numerical solution and so is restricted to uniform grids on smooth closed curves superconvergence is possible with special choices of the collocation points numerical experiments with a model problem confirm the convergence rates predicted by our theory

two fixed point theorems in complete random normed modules and their applications to backward stochastic equations abstract let f f p be a filtered probability space with a filtration f f t t 0 t satisfying the usual conditions and t a finite time l 0 f 0 the algebra of equivalence classes of f 0 measurable real valued random variables on l p f t r d the usual function space and l f 0 p f t r d the l 0 f 0 module generated by l p f t r d the usual backward stochastic equations b s e s are studied for their terminal conditions in l p f t r d motivated by the study of continuous time conditional mean conditional convex risk measure portfolio selection this paper for the first time formulates and studies a more general class of b s e s with their terminal conditions in l f 0 p f t r d the main results of this paper are to prove two fixed point theorems in complete random normed modules which are respectively the random generalizations of banach contraction mapping principle and browder kirk fixed point theorem and give their applications to the general class of b s e s

on the definition of the strategic stability of equilibria a new definition of strategic stability is shown to satisfy all of the requirements given by elon kohlberg and jean francois mertens 1986 the definition follows the general form of the original definition of kohlberg and mertens but rather than working with perturbations of the payoffs or strategy space works directly with perturbations to the best reply correspondence with the appropriate topology on this space of perturbations the resulting definition does satisfy all of the requirements given by kohlberg and mertens it is shown that one does not have much freedom in the topology one uses

some results concerning the generic continuity of set valued mappings abstract a theorem proved by fort in 1951 states that an upper semi continuous mapping from a baire space x into non empty compact subsets of a metric space is both lower and upper semi continuous at the points of a dense residual set of x in this paper we present several results concerning the generic continuity of set valued mappings first we demonstrate that generic continuity in the sense of baire category cannot be extended to almost everywhere continuity with respect to some measure then we show that baireness of the domain space is a necessary condition for fort s theorem to hold finally we show that fort s theorem holds without any compactness assumption on the images of the set valued mapping

a study of boundedness in probabilistic normed spaces probabilistic normed spaces have been redefined by alsina schweizer and sklar we give a detailed analysis of various boundedness notions for linear operators between such spaces and we study the relationship among them and also with the notion of continuity

on almost surely bounded semigroups of random linear operators menger proposed transferring the probabilistic notions of quantum mechanics to the underlying geometry following menger s idea the notion of random metric spaces is a random generalization of that of metric spaces and also plays an important role in the study of random operator equations the main difficulty of this article is to work out a skill to give a property peculiar to a special semigroup of random operators which is not involved in the classical case subsequently some random operators equations are studied in particular we discuss the schrodinger type random equation which considerably generalizes the corresponding result in the sense of skorohod

conditionally evenly convex sets and evenly quasi convex maps abstract evenly convex sets in a topological vector space are defined as the intersection of a family of open half spaces we introduce a generalization of this concept in the conditional framework and provide a generalized version of the bipolar theorem this notion is then applied to obtain the dual representation of conditionally evenly quasi convex maps which turns out to be a fundamental ingredient in the study of quasi convex dynamic risk measures

weak topologies for modules over rings of bounded random variables abstract in order to establish a functional analytic basis for representation theorems for conditional and multi period risk measures we study locally convex modules over the ring l g their topology is determined by seminorms as expected central mathematical tools of the analysis are hahn banach type and separation theorems which however have to be treated more carefully in the module case once a dual module is introduced one can establish a module version of the bipolar theorem we also prove the krein smulian as well as the alaoglu bourbaki theorem for modules for banach modules their reflexivity is characterized by a compactness criterium in a very weak topology

separation and duality in locally l 0 convex modules motivated by financial applications we study convex analysis for modules over the ordered ring l0 of random variables we establish a module analogue of locally convex vector spaces namely locally l0 convex modules in this context we prove hyperplane separation theorems we investigate continuity subdifferentiability and dual representations of fenchel moreau type for l0 convex functions from l0 modules into l0 several examples and applications are given

partial identification of functionals of the joint distribution of potential outcomes in this paper we present a systematic study of partial identification of two general classes of functionals of the joint distribution of two potential outcomes when a bivariate sample from the joint distribution is not available to the econometrician assuming the identification of the conditional marginal distributions of potential outcomes and the distribution of the covariate vector we show that the identified sets for functionals in both classes are intervals and provide conditions under which the identified sets point identify the true value of the functionals in addition we establish sufficient and necessary conditions for the covariate information to be informative in the sense of shrinking the identified sets we focus on the application of our general results to evaluating distributional treatment effects of a binary treatment in two commonly used frameworks in the literature for evaluating average treatment effects the selection on observables framework and a latent threshold crossing model we characterize the role of the propensity score in the selection on observables framework and the role of endogenous selection in the latent threshold crossing model examples of policy parameters that our results apply include the correlation coefficient between the potential outcomes many inequality measures of the distribution of treatment effects and median of the distribution of the individual treatment effect

farkas lemma in random locally convex modules and minkowski weyl type results in l0 f rn abstract in this paper we investigate the structure of a finitely generated l 0 positively homogeneous and l 0 convex cone in a random locally convex module first we establish a farkas type characterization for such a kind of cone then we can further give some minkowski weyl type results for the specific case when the random locally convex module is the space l 0 f r n of equivalence classes of r n valued random variables

the algebraic structure of finitely generated l0 f k modules and the helly theorem in random normed modules abstract let k be the scalar field of real numbers or complex numbers and l 0 f k the algebra of equivalence classes of k valued random variables defined on a probability space f p in this paper we first characterize the algebraic structure of finitely generated l 0 f k modules and then combining the recently developed separation theorem in random locally convex modules we prove the helly theorem in random normed modules with the countable concatenation property under the framework of random conjugate spaces at the same time a simple counterexample shows that it is necessary to require the countable concatenation property

a characterization for an l μ k topological module to admit enough canonical module homomorphisms let k be the scalar field of all real or complex numbers let a be a finite measure space and let l k be the algebra of the equivalence classes of all k valued measurable functions defined on a l k is a topological algebra over k when endowed with the topology of convergence locally in measure topological modules over this topological algebra l k briefly l k topological modules are an extensive class of topological modules which arise naturally in the course of the study of the theory of probabilistic normed spaces the purpose of this paper is to show that an arbitrary regular l k topological module admits enough canonical module homomorphisms if and only if all of its quasi free submodules of finite rank are complemented in the sense of topological modules

an equivalence relation between optimization problems connected with the well posedness let x be a hausdorff topological space and 23 be a family of real valued functions defined in x which are bounded from below taking an arbitrary f 3 we want to minimize f on x i e to find x ex such that f xo inf f x x e x this problem will be referred to as the minimization problem x f varying f in b we obtain a set of minimization problems we are interested in those functions from 23 for which the corresponding minimization problem has a unique solution and moreover some continuous dependence of this solution on the data of the problem exists such problems are called well posed the precise definition is given below suppose b is endowed with a uniform metric under which it is a complete metric space two important cases are i 8 consists of lower semicontinuous functions ii 23 consists of continuous functions in this case the following question is of some interest does the set f 23 x f is well posed contain a dense and g subset of 8 and from the point of view of the baire category is considered to be a big subset of d in a different setting including both the unconstrained and constrained case the answer to this question is positive and is given in l 7 l 12 14173 a possible reservation one may have regarding such results is mentioned by beer in 11 following kenderov 10 111 he considers the next set of constrained minimization problems cp a f 0 a c x a is compact f x r f is continuous and bounded x is tech complete it is observed in l that the reason for the topological bigness of the set mf e p af is well posed could be the fact that in cp there exist different couples giving the same minimization problem indeed let a f b g e cp a b f g but the restricti0ns f 1 a and g 1 b coincide then as

on the strongly essential components of nash equilibria of infinite n person games with quasiconcave payoffs abstract in this paper we introduce a concept of strongly essential sets of fixed points and establish the existence of the same in the perturbations of upper semicontinuous mappings with nonempty and compact values as an application we show that for infinite n person noncooperative games with quasiconcave payoffs there exists at least one strongly essential component of nash equilibria and which also contains at least one hyperstable set and one quasistable set

generic uniqueness of equilibrium in large crowding games a crowding game is a noncooperative game in which the payoff of each player depends only on the player s action and the size of the set of players choosing that particular action the larger the set the smaller the payoff finite n player crowding games often have multiple equilibria however a large crowding game generically has just one equilibrium and the equilibrium payoffs in such a game are always unique moreover the sets of equilibria of them replicas of a finite crowding game generically converge to a singleton asm tends to infinity this singleton consists of the unique equilibrium of the limit large crowding game this equilibrium generically has the following graph theoretic property the bipartite graph in which each player in the original finite crowding game is joined with all best response actions for copies of that player does not contain cycles

on the super approximation property of galerkin s method with finite elements for galerkin s method with finite elements as trial functions for strongly elliptic operator equations in the hilbert scaleh t the super approximation property and the optimal convergence rate are obtained by using the aubin nitsche lemma this applies in particular to spline collocation methods for a wide class of pseudodifferential equations

bounds on distributional treatment effect parameters using panel data with an application on job displacement abstract this paper develops new techniques to bound distributional treatment effect parameters that depend on the joint distribution of potential outcomes an object not identified by standard identifying assumptions such as selection on observables or even when treatment is randomly assigned i show that panel data and an additional assumption on the dependence between untreated potential outcomes for the treated group over time i provide more identifying power for distributional treatment effect parameters than existing bounds and ii provide a more plausible set of conditions than existing methods that obtain point identification i apply these bounds to study heterogeneity in the effect of job displacement during the great recession using standard techniques i find that workers who were displaced during the great recession lost on average 34 of their earnings relative to their counterfactual earnings had they not been displaced using the methods developed in the current paper i also show that the average effect masks substantial heterogeneity across workers

best possible bounds for the distribution of a sum a problem of kolmogorov recently in answer to a question of kolmogorov g d makarov obtained best possible bounds for the distribution function of the sumx y of two random variables x andy whose individual distribution functions fx andfy are fixed we show that these bounds follow directly from an inequality which has been known for some time the techniques we employ which are based on copulas and their properties yield an insightful proof of the fact that these bounds are best possible settle the question of equality and are computationally manageable furthermore they extend to binary operations other than addition and to higher dimensions

the law of large numbers for fuzzy variables under a general triangular norm extension principle this paper presents a law of large numbers for t noninteractive weakly noninteractive fuzzy variables when using a general t norm extension principle the result is that convergence to a crisp set is obtained for all archimediean t norm intersection operators this generalises a previous result of badard the result is proved by deriving it from a similar result for the law of large numbers for dependency bounds of sums of random variables dependency bounds are used when nothing is known of the joint distribution of two random variables apart from their marginals the bridge used to connect dependency bounds with fuzzy number operations is used to give a probabilistic interpretation of fuzzy number operations some numerical examples are presented and a new method of numerically calculating the sup t convolutions of fuzzy numbers is presented this new method generalises the well known result for the calculation of sup min convolutions in terms of the level sets of membership functions

stability of efficient sets continuity of mobile polarities whenever for every x e d such that a x afi xo for i 1 2 n one hash x fi x for each i pareto optimization is also referred to as uecror multi objecfiue or multi criteria optimization multi objective maximization problems 0 1 have been nowadays extended to the situations where f is a mapping from x to a set y equipped with a transitive relation 2 this relation is frequently called a preference consider first the image a f d of the constraint d an element y of a is called 3 efficient up to indifference or in the broad sense if for every y e a y z y implies y 5 yo it is said to be strongly z eficient or efficient in the narrow sense if for each y e a y 2 y implies y y the two notions coincide for antisymmetric transitive relations we shall be primarily concerned with efficient points in the broad sense the set of efficient points of a with respect to 7 will be denoted by max a max z a and that of strongly efficient points by max a an element of y is called an efficient value of 0 1 if it is an efficient point of f d an element x0 of d is a solution of 0 1 if f xo is an efficient point of f d we shall use the latter s to denote the set of solutions of 0 1

generic uniqueness of equilibrium points abstract let x be a nonempty convex and compact subset of normed linear space e respectively let x be a nonempty bounded closed and convex subset of banach space e and a be a nonempty convex and compact subset of x and f x x r be a given function the uniqueness of equilibrium point for equilibrium problem which is to find x x respectively x a such that f x y 0 for all y x respectively f x y 0 for all y a is studied with varying f respectively with both varying f and varying a the results show that most of equilibrium problems in the sense of baire category have unique equilibrium point

generic existence of lipschitzian solutions of optimal control problems without convexity assumptions the tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions in this paper we consider a large class of optimal control problems which is identified with a complete metric space of integrands without convexity assumptions and show that for a generic integrand the corresponding optimal control problem possesses a unique solution and this solution is lipschitzian

on the asymptotic convergence of collocation methods we prove quasioptimal and optimal order estimates in various sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines the analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications equations to which the analysis is applied include fredholm integral equations of the second kind certain first kind fredholm equations singular integral equations involving cauchy kernels a variety of integro differential equations and two point boundary value problems for ordinary differential equations the error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard galerkin methods we compare the collocation method with a standard galerkin method using splines of the same degree showing that the galerkin method is quasioptimal in a sobolev space of lower index and furnishes optimal order approximation for a range of sobolev indices containing and extending below that for the collocation method and so the standard galerkin method achieves higher rates of convergence

ekelandʼs variational principle for an l 0 valued function on a complete random metric space abstract motivated by the recent work on conditional risk measures this paper studies the ekeland s variational principle for a proper lower semicontinuous and lower bounded l 0 valued function where l 0 is the set of equivalence classes of extended real valued random variables on a probability space first we prove a general form of ekeland s variational principle for such a function defined on a complete random metric space then we give a more precise form of ekeland s variational principle for such a local function on a complete random normed module finally as applications we establish the bishop phelps theorem in a complete random normed module under the framework of random conjugate spaces

conditional lp spaces and the duality of modules over f algebras abstract motivated by dynamic asset pricing we extend the dual pairs theory of dieudonne 1942 and mackey 1945 to pairs of modules over a dedekind complete f algebra with multiplicative unit the main tools are a hahn banach theorem for modules of this kind a topology on the f algebra that has the special feature of coinciding with the norm topology when the algebra is a banach algebra and with the strong order topology of filipovic kupper and vogelpoth 2009 when the algebra of all random variables on a probability space g p is considered as a leading example we study in some detail the duality of conditional l p spaces

well posedness of a class of perturbed optimization problems in banach spaces abstract let x be a banach space and z a nonempty subset of x let j z r be a lower semicontinuous function bounded from below and p 1 this paper is concerned with the perturbed optimization problem of finding z 0 z such that x z 0 p j z 0 inf z z x z p j z which is denoted by min j x z the notions of the j strictly convex with respect to z and of the kadec with respect to z are introduced and used in the present paper it is proved that if x is a kadec banach space with respect to z and z is a closed relatively boundedly weakly compact subset then the set of all x x for which every minimizing sequence of the problem min j x z has a converging subsequence is a dense g subset of x z 0 where z 0 is the set of all points z z such that z is a solution of the problem min j z z if additionally p 1 and x is j strictly convex with respect to z then the set of all x x for which the problem min j x z is well posed is a dense g subset of x z 0

generic well posedness of minimization problems with mixed continuous constraints in this paper we study mathematical programming problems with mixed constraints in a banach space and show that most of the problems in the baire category sense are well posed our result is a generalization of a result of ioffe et al siam j optim 12 2001 461 478 obtained for finite dimensional banach spaces

the bishop phelps theorem in complete random normed modules endowed with the e λ topology abstract in this paper we adopt a new approach so that we can prove that the bishop phelps theorem in complete random normed modules still holds under the e topology which solves an open problem posed in t x guo y j yang ekeland s variational principle for an l 0 valued function on a complete random metric space j math anal appl 389 2012 1 14

stability in locally l0 convex modules and a conditional version of james compactness theorem abstract locally l 0 convex modules were introduced in filipovic et al 2009 10 as the analytic basis for the study of conditional risk measures later the algebra of conditional sets was introduced in drapeau et al 2016 8 in this paper we study locally l 0 convex modules and find exactly which subclass of locally l 0 convex modules can be identified with the class of locally convex vector spaces within the context of conditional set theory second we provide a version of the classical james theorem of characterization of weak compactness for conditional banach spaces finally we state a conditional version of the fatou and lebesgue properties for conditional convex risk measures and as application of the developed theory we establish a version of the so called jouini schachermayer touzi theorem for robust representation of conditional convex risk measures defined on a l type module

lipschitz like property of an implicit multifunction and its applications abstract the aim of this work is twofold first we use the advanced tools of modern variational analysis and generalized differentiation to study the lipschitz like property of an implicit multifunction more explicitly new sufficient conditions in terms of the frechet coderivative and the normal mordukhovich coderivative of parametric multifunctions for this implicit multifunction to have the lipschitz like property at a given point are established then we derive sufficient conditions ensuring the lipschitz like property of an efficient solution map in parametric vector optimization problems by employing the above implicit multifunction results

generic uniqueness of area minimizing disks for extreme curves we show that for a generic nullhomotopic simple closed curve gamma in the boundary of a compact orientable mean convex 3 manifold m with h 2 m bf z 0 there is a unique area minimizing disk d embedded in m with partial d gamma we also show that the same is true for nullhomologous curves in the absolutely area minimizing surface case

a basic strict separation theorem in random locally convex modules abstract the central idea of this paper is to make full use of the recently developed theory of random conjugate spaces to establish a basic strict separation theorem that is universally suitable in an arbitrary random locally convex module a series of interesting corollaries of the basic theorem are also included

an extreme limit theorem for dependency bounds of normalized sums of random variables dependency bounds are lower and upper bounds on the probability distribution of functions of random variables when only their marginal distributions are known using the properties of t conjugate transforms we show that dependency bounds for normalized sums of random variables converge to step functions as the number of summands increases the step functions are positioned at points depending only on the extremes of the supports of the summands distribution functions

on essential sets and essential components of efficient solutions for vector optimization problems in this paper we first introduce the notions of an essential set and an essential component of the set of efficient solutions for continuous vector optimizations on a nonempty compact subset of a metric space then we show that for each of these vector optimizations each set of all efficient solutions corresponding to the same optimal values is essential basing on this result we give full characterizations of an essential point an essential set and an essential component respectively as an application we prove that for continuous quasiconvex vector optimization problems on a nonempty compact subset of a metric vector space each component of the set of efficient solutions is essential even though the efficient solution set is not connected

relations between some basic results derived from two kinds of topologies for a random locally convex module the purpose of this paper is to exhibit the relations between some basic results derived from the two kinds of topologies namely the e topology and the stronger locally l0 convex topology for a random locally convex module first we give an extremely simple proof of the known hahn banach extension theorem for l0 linear functions as well as its continuous variant then we give the relations between the hyperplane separation theorems in d filipovic m kupper n vogelpoth separation and duality in locally l0 convex modules j funct anal 256 2009 3996 4029 and a basic strict separation theorem in t x guo h x xiao x x chen a basic strict separation theorem in random locally convex modules nonlinear anal 71 2009 3794 3804 in the process we also obtain a very useful fact that a random locally convex module with the countable concatenation property must have the same completeness under the two topologies as applications of the fact we prove that most of the previously established principal results of random conjugate spaces of random normed modules under the e topology are still valid under the locally l0 convex topology which considerably enriches financial applications of random normed modules

continuity properties of solutions of vector optimization this paper characterizes upper semicontinuity and lower semicontinuity of weakly efficient solution mapping sw and efficient solution mapping s on the space cm x of continuous m dimension vector real valued objective functions on a nonempty compact set x as applications some more interesting results about the properties of weakly efficient solutions and efficient solutions are obtained such as the upper semicontinuity of s at f in cm x implies the closedness and connectedness of the set s f of efficient solutions any essential weakly efficient solution of f can be approximated by a sequence of efficient solutions of f i e ew f s f and in the sense of baire categories most of vector optimization problems in cm x for efficient solutions are essential and so on

the james theorem in complete random normed modules the purpose of this paper is to generalize the classical james theorem characterizing the reflexivity of banach spaces to one characterizing the random reflexivity of complete random normed modules

pseudo lipschitz property of linear semi infinite vector optimization problems this paper is devoted to the study of the pseudo lipschitz property of pareto solution map for the parametric linear semi infinite vector optimization problem lsvo we establish new sufficient conditions for the pseudo lipschitz property of the pareto solution map of lsvo under continuous perturbations of the right hand side of the constraints and linear perturbations of the objective function examples are given to illustrate the results obtained

partial identification of the treatment effect distribution and its functionals abstract in the treatment effect problem the available information is on the marginal distributions of potential outcomes but not on their joint distribution the only point identified functional of the treatment effect distribution is its average the average treatment effect ate quantiles and other functionals of the distribution of treatment effect are only partially identified bounds on a single quantile and on the cumulative distribution function c d f in a single point are sharp makarov bounds we show that bounds on functionals of the quantile process that use makarov bounds are not sharp because the makarov bounds are pointwise but not uniformly sharp this allows us to propose improved bounds on functionals of the c d f as an intermediate result we find that the makarov bounds on the region that contains the c d f of the treatment effect distribution in a finite number of points can be improved we provide numerical illustrations throughout the paper permitting a clear visualization of how the method works

the mean ergodic theorem on random reflexive random normed modules in this paper we prove a mean ergodic theorem for an almost surely power bounded random linear operator on a random reflexive random normed module which general izes and improves several known important results

a unified approach to time consistency of dynamic risk measures and dynamic performance measures in discrete time in this paper we provide a flexible framework allowing for a unified study of time consistency of risk measures and performance measures also known as acceptability indices the proposed framework not only integrates existing forms of time consistency but also provides a comprehensive toolbox for analysis and synthesis of the concept of time consistency in decision making in particular it allows for in depth comparative analysis of most of the existing types of time consistency a feat that has not been possible before and which is done in the companion paper by the authors in our approach the time consistency is studied for a large class of maps that are postulated to satisfy only two properties monotonicity and locality we call these maps lm measures the time consistency is defined in terms of an update rule the form of the update rule introduced here is novel and is perfectly suited for developing the unifying framework that is worked out in this paper as an illustration of the applicabi

stability of semi infinite vector optimization problems without compact constraints this paper is devoted to the continuity of solution maps for perturbation semi infinite vector optimization problems without compact constraint sets the sufficient conditions for lower semicontinuity and upper semicontinuity of solution maps under functional perturbations of both objective functions and constraint sets are established some examples are given to analyze the assumptions in the main result

the algebra of conditional sets and the concepts of conditional topology and compactness abstract the concepts of a conditional set a conditional inclusion relation and a conditional cartesian product are introduced the resulting conditional set theory is sufficiently rich in order to construct a conditional topology a conditional real and functional analysis indicating the possibility of a mathematical discourse based on conditional sets it is proved that the conditional power set is a complete boolean algebra and a conditional version of the axiom of choice the ultrafilter lemma tychonoff s theorem the borel lebesgue theorem the hahn banach theorem the banach alaoglu theorem and the krein smulian theorem are shown

hilbert a modules abstract we consider real pre hilbert modules h on archimedean f algebras a with unit e we provide conditions on a and h such that a riesz representation theorem for bounded continuous a linear operators holds