Quotient normed cones

Given a normed cone (X, p) and a subconeY, we construct and study the quotient normed cone (X/Y,p) generated byY. In particular we characterize the bicompleteness of (X/Y, ‖·‖ ^p ,p) in terms of the bicompleteness of (X, p), and prove that the dual quotient cone ((X/Y)^*, || · ‖·‖^p,p) can be identified as a distinguished subcone of the dual cone (X ^*, || · ||_{p, u}). Furthermore, some parts of the theory are presented in the general setting of the spaceCL(X, Y) of all continuous linear mappings from a normed cone (X, p) to a normed cone (Y, q), extending several well-known results related to open continuous linear mappings between normed linear spaces.     

The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem in complete random normed modules to stratification structure

Let (Ω,A,μ) be a probability space, K the scalar field R of real numbers or C of complex numbers,and (S,X) a random normed space over K with base (ω,A,μ). Denote the support of (S,X) by E, namely E is the essential supremum of the set {A ∈ A: there exists an element p in S such that X _p (ω) > 0 for almost all ω in A}. In this paper, Banach-Alaoglu theorem in a random normed space is first established as follows: The random closed unit ball S ^*(1) = {f ∈ S ^*: X ^* _f ⩽ 1} of the random conjugate space (S ^*,X ^*) of (S,X) is compact under the random weak star topology on (S ^*,X ^*) iff E∩A=: {E∩A | A ∈ A} is essentially purely μ-atomic (namely, there exists a disjoint family {A _n : n ∈ N} of at most countably many μ-atoms from E ∩ A such that E = ∪ _n=1^∞ A _n and for each element F in E ∩ A, there is an H in the σ-algebra generated by {A _n : n ∈ N} satisfying μ(FΔH) = 0), whose proof forces us to provide a key topological skill, and thus is much more involved than the corresponding classical case. Further, Banach-Bourbaki-Kakutani-Šmulian (briefly, BBKS) theorem in a complete random normed module is established as follows: If (S,X) is a complete random normed module, then the random closed unit ball S(1) = {p ∈ S: X _p ⩽ 1} of (S,X) is compact under the random weak topology on (S,X) iff both (S,X) is random reflexive and E ∩ A is essentially purely μ-atomic. Our recent work shows that the famous classical James theorem still holds for an arbitrary complete random normed module, namely a complete random normed module is random reflexive iff the random norm of an arbitrary almost surely bounded random linear functional on it is attainable on its random closed unit ball, but this paper shows that the classical Banach-Alaoglu theorem and BBKS theorem do not hold universally for complete random normed modules unless they possess extremely simple stratification structure, namely their supports are essentially purely μ-atomic. Combining the James theorem and BBKS theorem in complete random normed modules leads directly to an interesting phenomenum: there exist many famous classical propositions that are mutually equivalent in the case of Banach spaces, some of which remain to be mutually equivalent in the context of arbitrary complete random normed modules, whereas the other of which are no longer equivalent to another in the context of arbitrary complete random normed modules unless the random normed modules in question possess extremely simple stratification structure. Such a phenomenum is, for the first time, discovered in the course of the development of random metric theory.     

Dual operator norms and the spectra of matrices

Let ∥·∥ be an operator norm and ∥·∥^D its dual. Then it is shown that ∥A∥^D? ∑|λ_i(A)|, where λ_i(A) are the eigenvalues of A, holds for all matrices A if and only if ∥·∥ is the operator norm subordinate to a Euclidian vector norm.     

Extensions and new observations of Tychonoff,Stone-Weierstrass theorems,compactifications and the realcompactification

An extension of the Tychonoff theorem is obtained in characterizing a compact space by the nets and the images induced by any family of continuous functions on it. The idea of this extension is applied to get a new process and new observations of compactifications and the realcompactification. Finally, a sufficient and necessary condition of a vector sublattice or a subalgebra of C¹(X) to be dense in (C¹(X),∥·∥) is provided in terms of the nets in X induced by C¹(X), where C¹(X) is the space of all bounded real continuous functions on a topological space X with pointwise ordering, and ∥·∥ is the supremum norm.     

The bicompletion of an asymmetric normed linear space

A biBanach space is an asymmetric normed linear space (X,‖·‖) such that the normed linear space (X,‖·‖^s) is a Banach space, where ‖x‖^s= max {‖x‖,‖-x‖} for all x∈X. We prove that each asymmetric normed linear space (X,‖·‖) is isometrically isomorphic to a dense subspace of a biBanach space (Y,‖·‖_Y). Furthermore the space (Y,‖·‖_Y) is unique (up to isometric isomorphism). This revised version was published online in August 2006 with corrections to the Cover Date.     

Stability of the solutions of one-sided variational problems with applications to the theory of plasticity

In the paper one investigates the one-sided variational problem of the form $$/u - u_0 / = min, u \in M,$$ where ¦·¦ is the norm in some Hilbert space H₀, ? is a nonempty convex set, closed in the metric of H₀, and u₀ is a given element of this space. The fundamental results are: 1) The solution of the problem (1) is stable relative to small perturbations of the data of this problem: the element u₀, the norm ¦·¦, and the set ? the concept of small perturbation is precisely formulated. 2) Assume that the set ? is defined by the formula where g is an element of some Hilbert space containing the space H₀, ||| · ||| is some seminorm, and a is a positive constant. Let H⁽ⁿ⁾ be a subspace of the space H₀ on all of whose elements the seminorm ||| · ||| is finite. If u_n is the approximate solution of the problem (1), obtained as the solution of the problem ¦u?u₀¦=min, u_n ∈ M ∩ H⁽ⁿ⁾, then ¦u_*?u_n¦=0 (e_n(u_*)), where u_* is the exact solution of the problem (1), while e_n(u_*) is the best approximation of u_* by the elements of the subspace H⁽ⁿ⁾. The given results are used in a series of problems regarding the elastoplastic state according to the Saint-Venant-Mises theory; one assume that for these problems the Haar-Karman variational principle holds.     

On the Density of Positive Proper Efficient Points in a Normed Space

In the context of vector optimization and generalizing cones with bounded bases, we introduce and study quasi-Bishop-Phelps cones in a normed space X. A dual concept is also presented for the dual space X*. Given a convex subset A of a normed space X partially ordered by a closed convex cone S with a base, we show that, if A is weakly compact, then positive proper efficient points are sequentially weak dense in the set E(A, S) of efficient points of A; in particular, the connotation weak dense in the above can be replaced by the connotation norm dense if S is a quasi-Bishop-Phelps cone. Dually, for a convex subset of X* partially ordered by the dual cone S ⁺, we establish some density results of positive weak* efficient elements of A in E(A, S ⁺).     

Directions of uniform rotundity in direct sums of normed spaces

Let {(X_i,|| · || _i)}_{i ? I},\{(X_i,\left \| {\cdot } \right \| _{i})\}_{i\in I}, be an arbitrary family of normed spaces and let (E,|| · || _E)(E,\left \| {\cdot } \right \| _{E}) be a monotonic normed space of real functions on the set I that is an ideal in \Bbb R^I{\Bbb R}^I. We prove a sufficient condition for the direct sum space E(X_i) to be uniformly rotund in a direction. We show that this condition is also necessary for E=l_￥E=\ell _{\infty }, and it is not necessary for E=l₁E=\ell _1. When E is either uniformly rotund in every direction and has compact order intervals, or weakly uniformly rotund respect to its evaluation functionals, we reestablish as a corollary the result that reads: E(X_i)E(X_i) is uniformly rotund in every direction if and only if so are all the X_i.     

Hyperfinite Dimensional Representations of Spaces and Algebras of Measures

Let X be a locally compact topological space and (X, E, X_ω) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set X_ω ⊆ X, such that all internal subsets of X_ω are relatively compact in the induced topology and X is homeomorphic to the quotient X_ω/E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function X ? ^*\Bbb CX \rightarrow {}{^{\ast}{\Bbb C}} . The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient M_w(X)/M₀(X){\cal M}_{\omega}(X)/{\cal M}_0(X) , for certain external subspaces M₀(X), M_w(X){\cal M}_0(X), {\cal M}_{\omega}(X) of the hyperfinite dimensional Banach space ^*\Bbb C^X{}{^{\ast}{\Bbb C}}^X , with the norm ‖f‖₁ = ∑_{x ∈ X} |f(x)|. If additionally X = G is a hyperfinite group, X_ω = G_ω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G₀ of G_ω, and G is isomorphic to the locally compact group G_ω/G₀, then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and M_w(G)/M₀(G){\cal M}_{\omega}(G)/{\cal M}_0(G) are isometrically isomorphic as Banach algebras.     

Fractional Covers for Convolution Products

A non-zero vector-valued sequence u ∈ ?^q(X′) is a cover for a subset M of ?_P(X) if, for some 0 < α ≤ 1, ∥u * h∥ _∞ ≥ α ∥u∥_q ∥h∥_p for all h ∈ M. Covers of ?¹ = ?¹(R) are important in worst case system identification in ?¹ and in the reconstruction of elements in a normed space from corrupted functional values. We investigate the existence of covers for certain naturally occurring subspaces of ?^p(X). We show that there exist finitely supported covers for some subspaces, and obtain lower bounds for their ’lengths’. We also obtain similar results for covers associated with convolution products for spaces of measurable vector-valued functions defined on the positive real axis.     

The Choquet integral representation in the affine vector-valued case

LetX be any compact convex subset of a locally convex Hausdorff space andE be a complex Banach space. We denote byA(X, E) the space of all continuous and affineE-valued functions defined onX. In this paper we prove thatX is a Choquet simplex if and only if the dual ofA(X, E) is isometrically isomorphic by a selection map toM _m (X, E*), the space ofE*-valued,w*-regular boundary measures onX. This extends and strengthens a result of G. M. Ustinov. To do this we show that for any compact convex setX, each element of the dual ofA(X, E) can be represented by a measure inM _m (X, E*) with the same norm, and this representation is unique if and only ifX is a Choquet simplex. We also prove that ifX is metrizable andE is separable then there exists a selection map from the unit ball of the dual ofA(X, E) into the unit ball ofM _m (X, E*) which is weak* to weak*-Borel measurable.This work will constitute a portion of the author's Ph.D. Thesis at the University of Illinois.     

Completely linear functionals in partially ordered spaces

For an arbitrary normed space X the set (X^**) in X^** is introduced. It is proved that if X is a KN-lineal then ^*=(X^**), where ^* is the Nakano dual to the Banach dual X^*. By the same token X* is not in any way related with any partial ordering in the KN-lineal X.Translated from Matematicheskie Zametki, Vol. 8, No. 2, pp. 187–195, August, 1970.     

Certain numerical characteristics of KN-lineals

We study the properties of certain numerical characteristics in a normed lattice that characterize its conjugate space. A typical result is as follows: let X be a K_σN-space or a KB-lineal. If for every sequence {x_n} ? X of pairwise disjoint positive elements with norms not exceeding 1 we have $$\mathop {\underline {\lim } }\limits_{n \to \infty } \frac{1}{n} x_1 \vee x_2 \vee \ldots \vee x_n \parallel = 0,$$ then all the odd conjugate spaces X^*, X^***, ... are KB-spaces.     

Subseries convergence in the space of functions of bounded φ-variation

Let there be given two F-spaces (X, ∥ ∥*), (Y, ∥ ∥*), X ? Y. A series Σ ₁ ^∞ x_i of elements in X is said to have property O(X, Y) (in the case when X = Y, ∥ ∥ = ∥ ∥*-property O) if perfect boundedness of this series in (X, ∥ ∥) implies its perfect (subseries) convergence in ( Y, ∥ ∥*). Conditions are determined for a series of elements in the space of functions of bounded φ-variation to have property O(X, Y). These conditions are rephrased for the case when convergence with respect to the norm ∥∥ _? ^v is replaced by modular convergence generated by φ-variation.     

Analogs of the Luzin-Danzhu and Cantor-Lebesgue theorems for double trigonometric series

Let ∥ · ∥ be some norm in R², Γ be the unit sphere induced in R² by this norm, and {Aj} a sequence of disjoint subsets of R₊ such that if ν ε A_j, then ν · Γ ∩ Z^N ≠ Ø. For series of the form $$\sum\nolimits_{j = 1}^\infty {} \sum\nolimits_{\parallel n\parallel \in A_j } {c_n e^{2\pi _i (n_1 x_1 + n_2 x_2 )} } $$ analogs of the Luzin-Danzhu and Cantor-Lebesgue theorems are established.     

Cantor-Lebesgue theorem for double trignometric series

Let ∥·∥ be a norm in R² and let γ be the unit sphere induced by this norm. We call a segment joining points x,y ε R² rational if (x₁ ? y₁)/(x₂ ? y₂) or (x₂ ? y₂)/(x₁ ? y₁) is a rational number. Let γ be a convex curve containing no rational segments. Satisfaction of the condition $$T_\nu (x) = \sum\nolimits_{\parallel n\parallel = \nu } {c_n e^{2\pi i(n_1 x_1 + n_2 x_2 )} } \to 0(\nu \to \infty )$$ in measure on the set e? [- 1/2,1/2)×[- 1/2, 1/2) =T² of positive planar measure implies ∥T _v ∥L₄ (T²) → 0(v → ∞). if, however, γ contains a rational segment, then there exist a sequence of polynomials {T _v } and a set E ? T², ¦E¦ > 0, such that T _v (x) → 0(v → ∞) on E; however, ¦c_n¦ ? 0 for ∥n∥ → ∞.     

A Grothendieck-type theorem for the space of totally measurable functions

Let Σ be a σ-algebra of subsets of a non-empty set Ω. Let X be a real Banach space and let X^* stand for the Banach dual of X. Let B(Σ, X) be the Banach space of Σ-totally measurable functions f: Ω → X, and let B(Σ, X)^* and B(Σ, X)^** denote the Banach dual and the Banach bidual of B(Σ, X) respectively. Let bvca(Σ, X^*) denote the Banach space of all countably additive vector measures ν: Σ → X^* of bounded variation. We prove a form of generalized Vitali-Hahn-Saks theorem saying that relative σ(bvca(Σ, X^*), B(Σ, X))-sequential compactness in bvca(Σ, X^*) implies uniform countable additivity. We derive that if X reflexive, then every relatively σ(B(Σ, X)^*, B(Σ, X))-sequentially compact subset of B(Σ, X)_c^~ (= the σ-order continuous dual of B(Σ, X)) is relatively σ(B(Σ, X)^*, B(Σ, X)^**)-sequentially compact. As a consequence, we obtain a Grothendieck type theorem saying that σ(B(Σ, X)^*, B(Σ, X))-convergent sequences in B(Σ, X)_c^~ are σ(B(Σ, X)^*, B(Σ, X)^**)-convergent.     

Vanishing theorem for sheaves of microfunctions at the boundary on cr–manifolds

Let X be a complex analytic manifold. Consider S?M?Xreal analytic submonifolds with codium ^R _MS=1,and let ω be a connected component of M\S. Let p∈S X_MT_M ^*X where T^* _Xdenotes the conormal bundle to M in X, and denote by ν(p) the complex radial Euler field at p. Denote by μ*(Ox) (for * = M, ω) the microlocalization of the sheaf of holomorphic functions along *.

Under the assumption dim^R(T_pT_M ^*X? ν(p)) = 1, a theorem of vanishing for the cohomology groups H^jμM(Ox)p is proved in [K-S 1, Prop. 11.3.1], j being related to the number of positive and negative eigenvalue for the Levi form of M.

Under the hypothesis dim^R(T_pT_S ^*X∩ν(p))=1, a similar result is proved here for the cohomology groups of the complex of microfunctions at the boundary μω(Ox).Stating this result in terms of regularity at the boundary for CR–hyperfunctions a local Bochner–type theorem is then obtained.     

Green’s relations and regularity for semigroups of transformations that preserve order and a double direction equivalence

Let T _X be the full transformation semigroup on a set X,

On the Solution Existence of Generalized Quasivariational Inequalities with Discontinuous Multifunctions

We study the following generalized quasivariational inequality problem: given a closed convex set X in a normed space E with the dual E ^*, a multifunction and a multifunction Γ:X→2 ^X , find a point such that , . We prove some existence theorems in which Φ may be discontinuous, X may be unbounded, and Γ is not assumed to be Hausdorff lower semicontinuous. The authors express their sincere gratitude to the referees for helpful suggestions and comments. This research was partially supported by a grant from the National Science Council of Taiwan, ROC. B.T. Kien was on leave from National University of Civil Engineering, Hanoi, Vietnam.