Infinite-dimensional Normed Linear Space Research Articles

On the Existence and Characterization of the Maxent Distribution Under General Moment Inequality Constraints

A broad set of sufficient conditions that guarantees the existence of the maximum entropy (maxent) distribution consistent with specified bounds on certain generalized moments is derived. Most results in the literature are either focused on the minimum cross-entropy distribution or apply only to distributions with a bounded-volume support or address only equality constraints. The results of this work hold for general moment inequality constraints for probability distributions with possibly unbounded support, and the technical conditions are explicitly on the underlying generalized moment functions. An analytical characterization of the maxent distribution is also derived using results from the theory of constrained optimization in infinite-dimensional normed linear spaces. Several auxiliary results of independent interest pertaining to certain properties of convex coercive functions are also presented.

A note on the topological transversality theorem for acyclic maps

A new topological transversality theorem is presented for acyclic maps. The analysis relies on Urysohn’s Lemma and the fact that the unit sphere is contractible in infinite dimensional normed linear spaces.

Asymptotically isometric copies of c0 and l1 in certain Banach spaces

If X is a separable Banach space, then X ∗ contains an asymptotically isometric copy of l 1 if and only if there exists a quotient space of X which is asymptotically isometric to c 0. If X is an infinite-dimensional normed linear space and Y is any Banach space containing an asymptotically isometric copy of c 0, then L( X, Y) contains an isometric copy of l ∞. If X and Y are two infinite-dimensional Banach spaces and Y contains an asymptotically isometric copy of c 0, then K w ∗ (X ∗,Y) contains a complemented asymptotically isometric copy of c 0.

Normal families of holomorphic functions and mappings on a Banach space

The authors lay the foundations for the study of normal families of holomorphic functions and mappings on an infinite-dimensional normed linear space. Characterizations of normal families, in terms of value distribution, spherical derivatives, and other geometric properties are derived. Montel-type theorems are established. A number of different topologies on spaces of holomorphic mappings are considered. Theorems about normal families are formulated and proved in the language of these various topologies. Normal functions are also introduced. Characterizations in terms of automorphisms and also in terms of invariant derivatives are presented.

The unit ball of every infinite-dimensional normed linear space contains a (1+ɛ)-separated sequence

The unit ball of every infinite-dimensional normed linear space contains a (1+ɛ)-separated sequence

Gaps between Spheres in Normed Linear Spaces

AbstractThe geometric notions of a gap and gap points between “concentric” spheres in a normed linear space are introduced and studied. The existence of gap points characterizes finitedimensional spaces. General conditions are given under which an infinite-dimensional normed linear space admits concentric spheres such that both these spheres and their dual spheres fail to have gap points.

Cells and Cellularity in Infinite-Dimensional Normed Linear Spaces

Certain concepts such as cells, cellular sets, point-like sets, and decomposition spaces are studied and related in normed linear spaces. The relationships between these concepts in general resemble somewhat the corresponding relationships in Euclidean space. There are certain topological properties in Euclidean n-space which can be conveniently studied as properties in normed linear spaces. In this paper con, cepts such as open and closed cells, cellular sets, point-like sets, and decomposition spaces are studied and related. Many, but not all, of the relationships between these concepts in infinite-dimensional normed linear spaces resemble the corresponding relationships in finite-dimensional spaces. Throughout the paper, E will denote an arbitrary normed linear space, and 0 will represent the zero element of E. For any positive real number r and any x ,E E let B r(x) =y 6 E: llx Y|| < r} and Sr(x) -y e E: l|x Y|| = r}. For convenience let Br Br(0) and Sr = Sr(0). 1. Tame cells. A closed subset C of E is a cell in E if there exists a homeomorphism from the pair (B1, SI) onto the pair (C, Bd C). C is tame if there exists a homeomorphism from E onto itself taking C onto Bl. A closed subset K of E\Int C is a collar of C if there exists a homeomorphism h from the triple (B1; B2 $SI) onto the triple (K u C; C, Bd (K u C)). Lemma 1.1. Let C be a cell in E, and let / be a homeomorphism /rom the pair (BP, S onto the pair (C, Bd C). Then there exists a homeomorphism h from E onto itself such that h B = fIB V2 ./2 Presented to the Society, April 20, 1968; received by the editors September 13, 1971 and, in revised form, February 23, 1972. AMS (MOS) subject classifications (1970). Primary 57A17, 57A60; Secondary 57A20.

Cells and cellularity in infinite-dimensional normed linear spaces

Certain concepts such as cells, cellular sets, point-like sets, and decomposition spaces are studied and related in normed linear spaces. The relationships between these concepts in general resemble somewhat the corresponding relationships in Euclidean space.

Stable homeomorphisms on infinite-dimensional normed linear spaces.

R. Y. T. Wong has recently shown that all homeomorphisms on a connected manifold modeled on infinite-dimensional separable Hilbert space are stable. In this paper we establish the stability of all homeomorphisms on a normed linear space E E such that E E is homeomorphic to the countable infinite product of copies of itself. The relationship between stability of homeomorphisms and a strong annulus conjecture is demonstrated and used to show that stability of all homeomorphisms on a normed linear space E E implies stability of all homeomorphisms on a connected manifold modeled on E E , and that in such a manifold collared E E -cells are tame.

Annulus Conjecture and Stability of Homeomorphisms in Infinite-Dimensional Normed Linear Spaces

If $E$ is an arbitrary infinite-dimensional normed linear space, it is shown that if all homeomorphisms of $E$ onto itself are stable, then the annulus conjecture is true for $E$. As a result, this confirms that the annulus conjecture for Hilbert space is true. A partial converse is that for those spaces $E$ which have some hyperplane homeomorphic to $E$, if the annulus conjecture is true for $E$ and if all homeomorphisms of $E$ onto itself are isotopic to the identity, then all homeomorphisms of $E$ onto itself are stable.

An infinite-dimensional Schoenflies theorem

Although important high-dimensional cases of the polyhedral Schoenflies theorem are yet unsettled (see [4] and [11]), there has been considerable success in the past ten years in generalizing the theorem, proved by Schoenflies in 1906, to the effect that every one-dimensional sphere embedded in the plane is tame. Of course Alexander's horned sphere and others show the necessity of a further condition to obtain a generalization to embeddings of S in E +1 for n> 1. Alexander proved tameness in the combinatorial and differential categories for n=2, M. H. A. Newman extended the result to include all finite dimensions in the combinatorial category [10] and Barry Mazur accomplished the same for the differential category [7]. In the topological category, Marston Morse showed that Mazur's result implies tameness of semilocally tame (bicollared) n-spheres in En+, [9] and at the same time Morton Brown gave an elegant proof independent of Mazur's result [3]. As might be expected, in infinite-dimensional normed linear spaces even the counterpart of the Jordan curve theorem does not hold. However, it is shown in this paper that with the bicollared condition one does get Jordan separation. In fact tameness is proved by a modification of Brown's technique. Hopefully this infinite-dimensional Schoenflies theorem will be as useful in infinite-dimensional topology as has been the case in finite dimensions. To give substance to this hope a number of applications are given including an annulus theorem, a contribution to the classification of convex sets and partial answers to questions raised by R. H. Bing and R. D. Anderson [2]. 1.0 DEFINITIONS. Unless otherwise stated, X denotes a topological space and E (with or without subscripts or superscripts) an infinite-dimensional normed linear space with the norm topology. Let Bt={xeFII xI ? t} and St={xeFII xI=t} for t > 0. If A c X, Bd A denotes the point-set boundary of A and 'A its interior. If h: (B2, S2) -> (A, Bd A) is a homeomorphism, then A is called a (closed) E-cell in X, h(B1) is a collared E-cell in X, collared by h and A is a tame E-cell if X= E and h extends to a space homeomorphism. On the other hand, if h: (B3 -B1, S3 u S1) -(A, Bd A) is a homeomorphism, A is called a (closed) E-annulus in X and h(S2) a bicollared E-sphere in X, bicollared by h. The prefix Ewill be omitted if no confusion is likely.

Annulus conjecture and stability of homeomorphisms in infinite-dimensional normed linear spaces

If E E is an arbitrary infinite-dimensional normed linear space, it is shown that if all homeomorphisms of E E onto itself are stable, then the annulus conjecture is true for E E . As a result, this confirms that the annulus conjecture for Hilbert space is true. A partial converse is that for those spaces E E which have some hyperplane homeomorphic to E E , if the annulus conjecture is true for E E and if all homeomorphisms of E E onto itself are isotopic to the identity, then all homeomorphisms of E E onto itself are stable.

A remark on the intersection of shells

There are many characterizations which distinguish finite-dimensional normed linear spaces from infinite-dimensional normed linear spaces. Perhaps the best known of these is the compactness of the unit ball. Recently, V. Klee [1] showed that in any infinite-dimensional normed linear space there exists a decreasing sequence of unbounded but linearly bounded closed convex sets whose intersection is empty. We will give here a somewhat similar condition which holds in all infinite-dimensional normed linear spaces but does not hold in any finite-dimensional space. We begin with the following terminology. U will denote the unit ball {x: ||x|| S 1 } and S the unit sphere {x: ||x|| = 1 }. A shell will be any set of the form {x: ri? IIx-xalI ? r2 for 0 1 such that (x+p U)Cr (-x+p U) CO, and S will be called finitely nonfiat if there exists a finite number of points {xi} '1 in S such that n(xj+pU) Co. A collection of sets has the finite intersection property if the intersection of the sets in any finite subcollection is not empty.

Operators with a Single Spectrum

Let X be an infinite dimensional normed linear space over the complex field Z. X will not be complete, in general, and its completion will be denoted by . If ℬ(X) is the algebra of all bounded linear operators in X then T ∈ ℬ(X) has a unique extension and . The resolvent set of T ∈ ℬ(X) is defined to beand the spectrum of T is the complement of ρ(T) in Z.

On a theorem of Klee

In 1953 and 1956, Klee [3], [4] proved that for E any infinitedimensional normed linear space and K any compact subset of E, E\K is homeomorphic to E. Klee's argument used sequences of bounded convex sets. In [5], Lin has some extensions of Klee's results using modifications of his methods. In this paper we give a short and elementary proof of a somewhat more general result' using only simple set-theoretic properties. A space S is said to be an a-space provided (1) S is an infinite-dimensional topological linear space, i.e., an infinite dimensional real vector space with a Hausdorff topology in which vector addition and scalar multiplication are jointly continuous, (2) S has a Schauder basis, i.e., a sequence {xi i>o of elements of S such that for each sES there is a unique sequence of scalars ai with s= aixi (convergence being in the topology of S) such that the function fi defined by fi(s) =ai is continuous for each i, and (3) there is a neighborhood U of the origin such that the elements xi I of the Schauder basis above are not in U. Henceforth, all spaces under discussion are to be a-spaces. For each i, let Mi denote the product of i copies of the reals with usual distance function di referring to distance between points, between a point and a set or between two sets. Let fi be as defined in condition (2) of the definition of an a-space and let gi be the map of S onto M, defined by gi(s) = (fi(s), f2(s), . . . , fi(s)). Since, by hypothesis, fi is continuous (for each i), then so is gj for each j. A set K CS is said to be projectible provided (1) K is closed, (2) for any pES\K, there is a j such that gj(p) is not an element of the closure of gj(K), and (3) there exist infnitely many i such that fi(K) is bounded above or below. The proof of the following lemma is trivial and is therefore omitted.

Two topological problems concerning infinite-dimensional normed linear spaces

Introduction. In the present work, we shall study the following two closely related topological problems concerning infinite-dimensional normed linear spaces. Problem I. Given a closed subset K of an infinite-dimensional normed linear space E, when is E K homeomorphic to E? Problem II. Given a closed subset K of an infinite-dimensional normed linear space E, when does there exist a periodic homeomorphism of E onto E with K as its set of fixed points? Using the fact that every nonreflexive Banach space contains a decreasing sequence of nonempty bounded closed convex subsets with empty intersection, Klee [11] proved that if K is a compact subset of a nonreflexive Banach space E, then E is homeomorphic to E K. Later [13], he showed that every infinite-dimensional normed linear space contains a decreasing sequence of unbounded but linear bounded (for the definition, see ?1) nonempty closed convex subsets with empty intersection. He used this result to prove that every infinite-dimensional normed linear space E is homeomorphic to E K where K is an arbitrary compact subset of E. As a consequence [13], if C is the unit cell of an infinite-dimensional normed linear space E, then there exists a homeomorphism i of C onto a closed half-space J in E such that i (Bd C) = Bd J. Klee [11] also proved that if E is either a nonreflexive strictly convexifiable Banach space or an infinite-dimensional lp-space, then Q is homeomorphic to Q U K where K is a compact convex subset of the bounding hyperplane of an open halfspace Q in E. Concerning the set of fixed points of a periodic homeomorphism of a topological space into itself, a classical result of Smith [19] states that if M is a finite-dimensional locally compact space, acyclic modp where p is

Absolute and Unconditional Convergence in Normed Linear Spaces

In 1933 Orlicz proved various results concerning unconditional convergence in Banach spaces (4), which were noted by Banach ((l), p. 240) who remarked that absolute and unconditional convergence are equivalent in finite dimensional Banach spaces, but that whether or not the two are non-equivalent in all infinite dimensional spaces was still an open question. MacPhail (3) gave a criterion for the equivalence of the two notions of convergence in a general Banach space and used it to prove non-equivalence in the spaces l1 and L1. In 1950 Dvoretzky and Rogers demonstrated the non-equivalence of the two types of convergence in any infinite dimensional normed linear space, using an elegant and instructive geometrical approach (2). The result has also been proved by a different method by Grothendieck (5).

Convex bodies and periodic homeomorphisms in Hilbert space

Introduction. We are going to discuss certain topological properties of infinite-dimensional normed linear spaces, the results being most complete for Hilbert space 'i. Let us begin by describing some results and questions of previous authors which are closely related to those of the present paper. From A. Tychonoff's fixed-point theorem ([31], 1935)(2) it follows that in the weak topology, the unit cell C= {x| |x|| < 1 } of 'i must have the fixedpoint property; that is, every weakly continuous map of C into itself admits at least one fixed point. In the norm topology, on the other hand, S. Kakutani ([13], 1943) described a homeomorphism without fixed points of C onto itself. He used this to show that the unit sphere S ={x x = 1 } is contractible and is a deformation retract of C. At the end of his paper, Kakutani raised several questions. Are any two of 'i, C, and S homeomorphic? Does C admit a periodic homeomorphism without fixed points? What is the situation in general Banach spaces? In partial answer to the last question, J. Dugundji ([7], 1951) proved that the unit cell of a normed linear space has the fixed point property only if the space is finite-dimensional. P. A. Smith had proved ([26], 1941) that each prime-period homeomorphism of Euclidean n-space En must have a fixed point and asked ([9, p. 259], 1949) whether 6i admits a period two homeomorphism without fixed points. 0. H. Keller ([14], 1931) proved that the infinite-dimensional compact convex subsets of are mutually homeomorphic and all homogeneous. W. A. Blankinship ([3], 1951) showed that if XC, and Cl X is compact, then i-i-X is contractible. In the present paper we answer the questions of Kakutani and Smith, strengthen the theorems of Keller and Blankinship, and establish some further topological properties of convex bodies and periodic homeomorphisms in 'i. The principal tools employed are (a) Mazur's homeomorphism [22] of the space (L') onto the space (L2); (b) the existence in every nonreflexive normed linear space of a decreasing sequence of bounded closed convex sets with empty intersection; (c) the existence in (L') of a one-parameter con-