a pieri type formula for isotropic flag manifolds we give the formula for multiplying a schubert class on an odd orthogonal or symplectic flag manifold by a special schubert class pulled back from the grassmannian of maximal isotropic subspaces this is also the formula for multiplying a type b respectively type c schubert polynomial by the schur p polynomial p m respectively the schur q polynomial q m geometric constructions and intermediate results allow us to ultimately deduce this formula from formulas for the classical flag manifold these intermediate results are concerned with the bruhat order of the infinite coxeter group b identities of the structure constants for the schubert basis of cohomology and intersections of schubert varieties we show that most of these identities follow from the pieri type formula and our analysis leads to a new partial order on the coxeter group b and formulas for many of these structure constants

a nonlinear analytic approach to the splitting theorem of jacobs de leeuw glicksberg abstract in this paper we study nonlinear analytic methods for linear contractive semigroups in banach spaces and apply them to the splitting theorem of jacobs de leeuw glicksberg using these results we obtain the extension of lin s proposition for a group of linear operators to a semigroup

random products of contractions in banach spaces we show that the random product of a finite number of w contractions converges weakly in all smooth reflexive banach spaces if one of the contractions is compact then the convergence is uniform let x 1 be a real banach space and t x x a linear operator recall that t is called a contraction if i tvi i v i for all vectors v in x we say that a contraction t satisfies condition w if whenever v is bounded and ivni itvni 0 it follows that the weak limn o vn tvn 0 the algebraic semigroup s generated by a possibly infinite set of contractions will be said to satisfy condition w if for any bounded sequence of vectors vn c x and a sequence of words wn from s such that i v n wnvni 0 the weak limn oo vn wnvn 0 now let t1 t2 be a sequence of contractions that satisfy condition w and let r be a self mapping of the set of natural numbers a random product of these contractions is the sequence sn n 1 2 defined by sn tr n tr n 1 tro 1 our purpose in this paper is to study the convergence properties of such products such a study is not only of intrinsic interest but is also motivated by applications to the numerical solution of partial differential equations 2 14 and linear inequalities 19 approximation theory 16 9 and computer tomography 18 11 we begin by noting several properties of contractions which satisfy condition w we shall call such contractions w contractions for short we then consider the algebraic semigroup generated by a finite number of w contractions and establish a weak convergence theorem for random products of these contractions in all smooth reflexive banach spaces theorem 1 this seems to be the first such result outside hilbert space it includes the hilbert space theorem due to amemiya and ando 1 we continue with a weak convergence result for semigroups generated by an infinite sequence of contractions theorem 2 finally we obtain a uniform convergence theorem for random received by the editors october 24 1988 and in revised form april 4 1989 1980 mathematics subject classification 1985 revision primary 47a05 secondary 47b05 65j10

finite element solution of a reynolds koiter coupled problem for the elastic journal bearing abstract the aim of this work is to present a numerical algorithm for solving a new elastohydrodynamic free boundary problem which models the lubricant pressure behaviour by means of reynolds equation jointly with the free boundary model of elrod adams for cavitation and the elastic deformation of the bearing with the koiter shell model a non local constraint on the pressure is imposed the proposed algorithm uncouples the hydrodynamic and elastic parts to achieve a fixed point for the two dimensional problem a combination of finite elements transport diffusion and duality methods is performed for the three dimensional elastic problem a finite element discretization is applied to a mixed formulation for the koiter model

halpern s iteration for bregman strongly nonexpansive mappings in reflexive banach spaces we investigate strong convergence for bregman strongly nonexpansive mappings by halpern s iteration in the framework of reflexive banach spaces using the obtained result convergence for a family of bregman strongly nonexpansive mappings is also discussed as applications we apply our main result to problems of finding zeros of maximal monotone operators and equilibrium problems in reflexive banach spaces

the alternating algorithm in a uniformly convex and uniformly smooth banach space abstract let x be a uniformly convex and uniformly smooth banach space assume that the m i i 1 r are closed linear subspaces of x p m i is the best approximation operator to the linear subspace m i and m m 1 m r we prove that if m is closed then the alternating algorithm given by repeated iterations of i p m r i p m r 1 i p m 1 applied to any x x converges to x p m x where p m is the best approximation operator to the linear subspace m this result in the case r 2 was proven in deutsch 4

right bregman nonexpansive operators in banach spaces abstract we introduce and study new classes of bregman nonexpansive operators in reflexive banach spaces these classes of operators are associated with the bregman distance induced by a convex function in particular we characterize sunny right quasi bregman nonexpansive retractions and as a consequence we show that the fixed point set of any right quasi bregman nonexpansive operator is a sunny right quasi bregman nonexpansive retract of the ambient banach space

a mortar finite element approach to lubricated contact problems abstract in this paper a new numerical approach based on mortar discretization of interface fields is proposed to solve lubricated contact problems between deformable solid bodies the reynolds approximation to the thin film fluid equations is used to describe the fluid behavior on the interface while the equations of small strain elasticity to which finite elements are applied are assumed to govern the solid body behavior the result of this approach is a monolithic scheme in which the fluid and solid mechanics are solved simultaneously without the need for iterative or staggered approaches between the fluid and solid phases key features of the approach are that the fluid film thickness directly related to the deformation of the solid phase is computed from a least squares projection based on dual basis functions as pioneered by wohlmuth b i wohlmuth discretization methods and iterative solvers based on domain decomposition springer verlag heidelberg 2001 the free boundary problem for the fluid phase corresponding to the boundary between lubricated and cavitated regions is regularized with a penalty method and thus resolved as part of the solution several numerical examples are given to demonstrate the approach

two strong convergence theorems for bregman strongly nonexpansive operators in reflexive banach spaces abstract we study the convergence of two iterative algorithms for finding common fixed points of finitely many bregman strongly nonexpansive operators in reflexive banach spaces both algorithms take into account possible computational errors we establish two strong convergence theorems and then apply them to the solution of convex feasibility variational inequality and equilibrium problems

iterative convergence to cesaro means for continuous pseudocontractive mappings abstract let f be a fixed strongly pseudocontractive mapping and t be a continuous pseudocontractive mapping with f t 0 the sequence z m is iteratively defined as follows z m t m f z m 1 t m 1 m 1 j 0 m t j z m m 0 where t m 0 1 satisfies the condition lim m t m 0 we prove that z m converges strongly to the unique solution p to some variational inequality in f t our results develop and complement the corresponding ones by matsushita daishi kuroiwa j math anal appl 294 2004 206 214 and shioji takahashi arch math 72 1999 354 359 and moore nnoli j math anal appl 260 2001 269 278 and su li appl math comput 181 2006 332 341 and song chen appl math comput 186 2007 1120 1128 and wangkeeree appl math comput 201 2008 239 249

gromov witten invariants on grassmannians we prove that any three point genus zero gromov witten invariant on a type a grassmannian is equal to a classical intersection number on a two step flag variety we also give symplectic and orthogonal analogues of this result in these cases the two step flag variety is replaced by a sub maximal isotropic grassmannian our theorems are applied in type a to formulate a conjectural quantum littlewood richardson rule and in the other classical lie types to obtain new proofs of the main structure theorems for the quantum cohomology of lagrangian and orthogonal grassmannians

a proximal projection method for finding zeros of set valued operators in this paper we study the convergence of an iterative algorithm for finding zeros with constraints for not necessarily monotone set valued operators in a reflexive banach space this algorithm which we call the proximal projection method is essentially a fixed point procedure and our convergence results are based on new generalizations of the browder s demiclosedness principle we show how the proximal projection method can be applied for solving ill posed variational inequalities and convex optimization problems with data given or computable by approximations only the convergence properties of the proximal projection method we establish also allow us to prove that the proximal point method with bregman distances whose convergence was known to occur for maximal monotone operators still converges when the operator involved in it is monotone with sequentially weakly closed graph

root theoretic young diagrams and schubert calculus planarity and the adjoint varieties abstract we study root theoretic young diagrams to investigate the existence of a lie type uniform and nonnegative combinatorial rule for schubert calculus we provide formulas for co adjoint varieties of classical lie type this is a simplest case after the co minuscule family where a rule has been proved by h thomas and the second author using work of r proctor our results build on earlier pieri type rules of p pragacz j ratajski and of a buch a kresch h tamvakis specifically our formula for og 2 2 n is the first complete rule for a case where diagrams are non planar yet the formulas possess both uniform and non uniform features using these classical type rules as well as results of p e chaput n perrin in the exceptional types we suggest a connection between polytopality of the set of nonzero schubert structure constants and planarity of the diagrams this is an addition to work of a klyachko and a knutson t tao on the grassmannian and of k purbhoo f sottile on cominuscule varieties where the diagrams are always planar

iterative approximation of fixed points of nonexpansive mappings let k be a nonempty closed convex subset of a real banach space e which has a uniformly g teaux differentiable norm and t k k be a nonexpansive mapping with f t x k tx x for a fixed 0 1 define s k k by sx 1 x tx x k assume that zt converges strongly to a fixed point z of t as t 0 where zt is the unique element of k which satisfies zt tu 1 t tzt for arbitrary u k let n be a real sequence in 0 1 which satisfies the following conditions c1 lim n 0 c2 n for arbitrary x0 k let the sequence xn be defined iteratively by xn 1 nu 1 n sxn then xn converges strongly to a fixed point of t

a combinatorial rule for co minuscule schubert calculus we prove a root system uniform concise combinatorial rule for schubert calculus of minuscule and cominuscule flag manifolds g p the latter are also known as compact hermitian symmetric spaces we connect this geometry to the poset combinatorics of proctor thereby giving a generalization of schutzenberger s jeu de taquin formulation of the littlewood richardson rule that computes the intersection numbers of grassmannian schubert varieties our proof introduces cominuscule recursions a general technique to relate the numbers for different lie types

the viscosity approximation process for quasi nonexpansive mappings in hilbert spaces in this paper we discuss the strong convergence of the viscosity approximation method in hilbert spaces relatively to the computation of fixed points of operators in the wide class of quasi nonexpansive mappings our convergence results improve previously known ones obtained for the class of nonexpansive mappings

bregman strongly nonexpansive operators in reflexive banach spaces abstract we present a detailed study of right and left bregman strongly nonexpansive operators in reflexive banach spaces we analyze in particular compositions and convex combinations of such operators and prove the convergence of the picard iterative method for operators of these types finally we use our results to approximate common zeros of maximal monotone mappings and solutions to convex feasibility problems

nonexpansive iterations in hyperbolic spaces one of the most active research areas in nonlinear functional analysis is the asymptotics of nonexpansive mappings most of the results however have been obtained in normed linear spaces it is natural therefore to try to develop a theory of nonexpansive iterations in more general infinite dimensional manifolds this is the purpose of the present paper more specifically we propose the class of hyperbolic spaces as an appropriate background for the study of operator theory in general and of iterative processes for nonexpansive mappings in particular this class of metric spaces which is defined in section 2 includes all normed linear spaces and hadamard manifolds as well as the hilbert ball and the cartesian product of hilbert balls in section 3 we introduce co accretive operators and their resolvents and present some of their properties in the fourth section we discuss the concept of uniform convexity for hyperbolic spaces section 5 is devoted to two new geometric properties of infinite dimensional banach spaces theorem 5 6 provides a characterization of banach spaces having these properties in terms of nonlinear accretive operators in sections 6 7 and 8 we study explicit implict and continuous iterations repectively using the same approach in all three sections we illustrate this common approach with the following special case let c be a closed convex subset of a hyperbolic space x p let t c f c be a nonexpansive mapping and let x be a point in c in order to study the iteration t x n 0 1 2 we set z 1 l n x 0 l n t x k clco zj j i l and d inf p y ty y e c the first step is to show that p x k lim p x t x n d this leads to the convergence m of lz when x is uniformly convex and to the weak convergence of z when x is a banach space which is reflexive and strictly convex when t is an averaged mapping we are also able to establish the following triple equality for all k 2 1

the recursive nature of cominuscule schubert calculus the necessary and sufficient horn inequalities which determine the non vanishing littlewood richardson coefficients in the cohomology of a grassmannian are recursive in that they are naturally indexed by non vanishing littlewood richardson coefficients on smaller grassmannians we show how non vanishing in the schubert calculus for cominuscule flag varieties is similarly recursive for these varieties the non vanishing of products of schubert classes is controlled by the non vanishing products on smaller cominuscule flag varieties in particular we show that the lists of schubert classes whose product is non zero naturally correspond to the integer points in the feasibility polytope which is defined by inequalities coming from non vanishing products of schubert classes on smaller cominuscule flag varieties while the grassmannian is cominuscule our necessary and sufficient inequalities are different than the classical horn inequalities

approximation of common fixed points of a countable family of nonexpansive mappings in a banach space in this paper to find a common fixed point of a family of nonexpansive mappings we introduce a halpern type iterative sequence then we prove that such a sequence converges strongly to a common fixed point of nonexpansive mappings moreover we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings and the problem of finding a zero of an accretive operator

approximation of zeros of inverse strongly monotone operators in banach spaces abstract in this paper we consider the projection algorithm studied by iiduka and takahashi 2008 10 for finding a solution of the variational inequality problem for an inverse strongly monotone operator in a banach space we first remark that under the assumptions imposed on the operator in their paper the iterative sequence converges weakly to a zero of the operator not just a solution of the variational inequality problem in our proof slightly modified from the original we do not assume the uniform smoothness of a space as was the case there finally using halpern s type method we modify this algorithm to obtain the strong convergence to a zero of an inverse strongly monotone operator which is nearest to the initial element of the algorithm in the sense of the bergman distance associated with the function 1 2 2

an algorithm for finding a common point of the solution set of a variational inequality and the fixed point set of a bregman relatively nonexpansive mapping in this paper we investigate an iterative scheme for finding a common point of the solution set of a variational inequality and the fixed point set of a bregman relatively nonexpansive mapping moreover a numerical example which shows the conclusion of our result is presented our results improve and generalize many known results in the current literature

iterating bregman retractions the notion of a bregman retraction of a closed convex set in euclidean space is introduced bregman retractions include backward bregman projections and forward bregman projections as well as their convex combinations and are thus quite flexible the main result on iterating bregman retractions unifies several convergence results on projection methods for solving convex feasibility problems it is also used to construct new sequential and parallel algorithms

an iterative method for equilibrium problems and a finite family of relatively nonexpansive mappings in a banach space abstract in this paper we introduce a new iterative method for finding a common element of the set of fixed points of a finite family of relatively nonexpansive mappings and the set of solutions of an equilibrium problem in uniformly convex and uniformly smooth banach spaces then we prove a strong convergence theorem by using the generalized projection

abel averages and holomorphically pseudo contractive maps in banach spaces abstract a class of maps in a complex banach space is studied which includes both unbounded linear operators and nonlinear holomorphic maps the defining property which we call pseudo contractivity is introduced by means of the abel averages of such maps we show that the studied maps are dissipative in the spirit of the classical lumer phillips theorem for pseudo contractive holomorphic maps we establish the power convergence of the abel averages to holomorphic retractions

strong convergence theorems for bregman totally quasi asymptotically nonexpansive mappings in reflexive banach spaces the purpose of this article is by using the shrinking projection method introduced by takahashi kubota and takeuchi to propose an iteration algorithm for bregman total quasi f asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of reflexive banach spaces as applications we apply our results to a system of equilibrium problems and zero point problem of maximal monotone mappings in reflexive banach spaces the results presented in the paper improve and extend the corresponding results of reich and sabach 2010 12 suantai et al 2012 13 nilsrakoo and saejung 2011 11 qin et al 2009 5 wang et al 2011 6 su et al 2010 7 martinez yanes and xu 2006 3 and others

characteristic inequalities of uniformly convex and uniformly smooth banach spaces abstract let x be a real banach space with dual x and moduli of convexity and smoothness x e and x respectively for 1 p denotes the duality mapping from x into 2 x with gauge function t p 1 and j p denotes an arbitrary selection for j p let a r 0 0 t is strictly increasing and there exists c 0 such that t c x t 2 and f r 0 0 t is convex nondecreasing and there exists k 0 such that k x it is proved that x is uniformly convex if and only if there is a a such that x y p x p p j p x y x y x y x and x is uniformly smooth if and only if there is a f such that x y p x p p j p x y x y x y x where for given function f f x y is defined by f x y p 0 1 x ty x p t f t y x ty x dt these inequalities which have various applications can be regarded as general banach space versions of the well known polarization identity occurring in hilbert spaces

strong convergence theorems for finitely many nonexpansive mappings abstract under the assumption that e is a reflexive banach space whose norm is uniformly geteaux differentiable and which has a weakly continuous duality mapping j with gauge function ceng cubiotti yao strong convergence theorems for finitely many nonexpansive mappings and applications nonlinear analysis 67 2007 1464 1473 introduced a new iterative scheme for a finite commuting family of nonexpansive mappings and proved strong convergence theorems about this iteration in this paper only under the hypothesis that e is a reflexive banach space which has a weakly continuous duality mapping j with gauge function and several control conditions about the iterative coefficient are removed we present a short and simple proof of the above theorem

a geometric littlewood richardson rule we describe a geometric littlewood richardson rule interpreted as deforming the intersection of two schubert varieties into the union of schubert varieties there are no restrictions on the base eld and all multiplicities arising are 1 this is important for applications this rule should be seen as a generalization of pieri s rule to arbitrary schubert classes by way of explicit homotopies it has straightforward bijections to other littlewood richardson rules such as tableaux and knutson and tao s puzzles this gives the rst geometric proof and interpretation of the littlewood richardson rule geometric consequences are described here and in v2 kv1 kv2 v3 for example the rule also has an interpretation in k theory suggested by buch which gives an extension of puzzles to k theory

the asymptotics of nonexpansive iterations abstract let d be a closed subset of a banach space x and t d d a nonexpansive mapping conditions are given on the space x for t to satisfy the following property of ergodic type t n x n converges either weakly or strongly to a vector v rather unexpectedly d is not assumed to be convex nor is i t assumed to satisfy any range condition in addition it is shown that v is the unique point of least norm in the closure of r i t if and only if i t satisfies a certain range condition at infinity several interesting applications to accretive operator and nonlinear semigroup theory are also included

littlewood richardson coefficients for reflection groups abstract in this paper we explicitly compute all littlewood richardson coefficients for semisimple and kac moody groups g that is the structure constants also known as the schubert structure constants of the cohomology algebra h g p c where p is a parabolic subgroup of g these coefficients are of importance in enumerative geometry algebraic combinatorics and representation theory our formula for the littlewood richardson coefficients is purely combinatorial and is given in terms of the cartan matrix and the weyl group of g however if some off diagonal entries of the cartan matrix are 0 or 1 the formula may contain negative summands on the other hand if the cartan matrix satisfies a i j a j i 4 for all i j then each summand in our formula is nonnegative that implies nonnegativity of all littlewood richardson coefficients we extend this and other results to the structure coefficients of the t equivariant cohomology of flag varieties g p and bott samelson varieties i g

on the quantum cohomology of adjoint varieties we study the quantum cohomology of quasi minuscule and quasi cominuscule homogeneous spaces the product of any two schubert cells does not involve powers of the quantum parameter higher than 2 with the help of the quantum to classical principle we give presentations of the quantum cohomology algebras these algebras are semi simple for adjoint non coadjoint varieties and some properties of the induced strange duality are shown

halpern s iteration in banach spaces abstract we prove strong convergence theorems for a sequence which is generated by halpern s iteration we also apply our result for finding zeros of an accretive operator our result improves the recent result of aoyama et al k aoyama y kimura w takahashi m toyoda approximation of common fixed points of a countable family of nonexpansive mappings in a banach space nonlinear anal 67 2007 2350 2360 by removing some assumptions on the parameters finally we discuss the new sufficient condition studied by song y song a new sufficient condition for the strong convergence of halpern type iterations appl math comput 198 2 2008 721 728 y song new strong convergence theorems for nonexpansive nonself mappings without boundary conditions comput math appl 56 6 2008 1473 1478 and correct the main result of song and chai y song x chai halpern iteration for firmly type nonexpansive mappings nonlinear anal 71 10 2009 4500 4506

bregman monotone optimization algorithms a broad class of optimization algorithms based on bregman distances in banach spaces is unified around the notion of bregman monotonicity a systematic investigation of this notion leads to a simplified analysis of numerous algorithms and to the development of a new class of parallel block iterative surrogate bregman projection schemes another key contribution is the introduction of a class of operators that is shown to be intrinsically tied to the notion of bregman monotonicity and to include the operators commonly found in bregman optimization methods special emphasis is placed on the viability of the algorithms and the importance of legendre functions in this regard various applications are discussed

restriction varieties and geometric branching rules abstract this paper develops a new method for studying the cohomology of orthogonal flag varieties restriction varieties are subvarieties of orthogonal flag varieties defined by rank conditions with respect to not necessarily isotropic flags they interpolate between schubert varieties in orthogonal flag varieties and the restrictions of general schubert varieties in ordinary flag varieties we give a positive geometric rule for calculating their cohomology classes obtaining a branching rule for schubert calculus for the inclusion of the orthogonal flag varieties in type a flag varieties our rule in addition to being an essential step in finding a littlewood richardson rule has applications to computing the moment polytopes of the inclusion of so n in su n the asymptotic of the restrictions of representations of sl n to so n and the classes of the moduli spaces of rank two vector bundles with fixed odd determinant on hyperelliptic curves furthermore for odd orthogonal flag varieties we obtain an algorithm for expressing a schubert cycle in terms of restrictions of schubert cycles of type a flag varieties thereby giving a geometric though not positive algorithm for multiplying any two schubert cycles

iterative methods for solving systems of variational inequalities in reflexive banach spaces we prove strong convergence theorems for three iterative algorithms which approximate solutions to systems of variational inequalities for mappings of monotone type all the theorems are set in reflexive banach spaces and take into account possible computational errors

a novel method for integrating first and second order differential equations in elastohydrodynamic lubrication for the solution of smooth isothermal line contact problems this paper contains details of recent developments in the analysis of elastohydrodynamic lubrication problems using the finite element method a steady state isothermal finite element formulation of the smooth line contact problem with newtonian lubricant behaviour is presented containing both first and second order formulations of the hydrodynamic equation previous problems with the limited range of applicability of both first and second order finite difference solutions have been overcome by summing both the first and second order equations weighted contributions application of the method to a range of problems unattainable by either single first or second order formulations is presented copyright 1999 john wiley sons ltd

convergence theorems for maximal monotone operators and fixed point problems in banach spaces our purpose in this paper is to prove strong convergence theorems for approximation of a common zero of a finite family of maximal monotone operators which is also a fixed point for a left bregman strongly relatively nonexpansive mapping in a reflexive banach space we also apply our results to approximation of solutions of nonlinear integral equations of hammerstein type in reflexive banach spaces

on the rate of convergence of iterated bregman projections and of the alternating algorithm abstract we study the alternating algorithm for the computation of the metric projection onto the closed sum of two closed subspaces in uniformly convex and uniformly smooth banach spaces for banach spaces which are convex and smooth of power type we exhibit a condition which implies linear convergence of this method we show these convergence results for iterates of bregman projections onto closed linear subspaces using an intimate connection between the metric projection onto a closed linear subspace and the bregman projection onto its annihilator we deduce the convergence rate results for the alternating algorithm from the corresponding results for the iterated bregman projection method

the nash blow up of a cominuscule schubert variety abstract we compute the nash blow up of a cominuscule schubert variety in particular we show that the nash blow up is algebraically isomorphic to another schubert variety of the same lie type as a consequence we give a new characterization of the smooth locus and for grassmannian schubert varieties determine when the nash blow up is a resolution of singularities we also study the induced torus action on the nash blow up and give a bijection between its torus fixed points and peterson translates on the schubert variety

existence and approximation of fixed points of firmly nonexpansive type mappings in banach spaces a class of nonlinear operators in banach spaces is proposed we call each operator in this class a firmly nonexpansive type mapping this class contains the classes of firmly nonexpansive mappings in hilbert spaces and resolvents of maximal monotone operators in banach spaces we study the existence and approximation of fixed points of firmly nonexpansive type mappings in banach spaces

a littlewood richardson rule for two step flag varieties this paper studies the geometry of one parameter specializations of subvarieties of grassmannians and two step flag varieties as a consequence we obtain a positive geometric rule for expressing the structure constants of the cohomology of two step flag varieties in terms of their schubert basis a corollary is a positive geometric rule for computing the structure constants of the small quantum cohomology of grassmannians we also obtain a positive geometric rule for computing the classes of subvarieties of grassmannians that arise as the projection of the intersection of two schubert varieties in a partial flag variety these rules have numerous applications to geometry representation theory and the theory of symmetric functions

james orthogonality and orthogonal decompositions of banach spaces abstract we establish decompositions of a uniformly convex and uniformly smooth banach space b and dual space b in the form b m j m and b m j m where m is an arbitrary subspace in b m is its annihilator subspace in b j b b and j b b are normalized duality mappings the sign denotes the james orthogonal summation in fact it is the direct sums of the corresponding subspaces and manifolds in a hilbert space h these representations coincide with the classical decomposition in a shape of direct sum of the subspace m and its orthogonal complement m h m m