Uniform Convexity in Factor and Conjugate Spaces

The notion of “exceptional family of elments (EFE)” plays a very important role in solving complementarity problems. It has been applied in finite dimensional spaces and Hilbert spaces by many authors. In this paper, by using the generalized projection defined by Alber, we extend this notion from Hilbert spaces to uniformly smooth and uniformly convex Banach spaces, and apply this extension to the study of nonlinear complementarity problems in Banach spaces.

We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a Radó–Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.

The notion of the shell of aHilbert spa ce opera tor, which is auseful genera liza tion (proposed by Wielandt) of the numerical range, is extended to operators in spaces with an indefinite inner product. For the most part, finite dimensional spaces are considered. Geometric properties of shells (convexity, boundedness, being a subset of a line, etc.) are described, as well as shells of operators in two dimensional indefinite inner product spaces. For normal operators, it is conjectured that the shell is convex and its closure is polyhedral; the conjecture is proved for indefinite inner product spaces of dimension at most three, and for finite dimensional inner product spaces with one positive eigenvalue.

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.

Linear algebra is considered an essential mathematical theory that has many engineering applications. While many theorem provers support linear spaces, they only consider finite dimensional spaces. In addition, available libraries only deal with real vectors, whereas complex vectors are extremely useful in many fields of engineering. In this paper, we propose a new linear space formalization which covers both finite and infinite dimensional complex vector spaces, implemented in HOL-Light. We give the definition of a linear space and prove many properties about its operations, e.g., addition and scalar multiplication. We also formalize a number of related fundamental concepts such as linearity, hermitian operation, self-adjoint, and inner product space. Using the developed linear algebra library, we were able to implement basic definitions about quantum mechanics and use them to verify a quantum beam splitter, an optical device that has many applications in quantum computing.

Given a signal, whether it is a discrete vector or a continuous function, one desires to write it in terms of simpler components. Typically, these components or “building blocks” form what is called a basis. A basis is an optimal set, containing the minimal number of elements needed to uniquely represent any signal in a given space. A frame can be thought of as a redundant basis, having more elements than needed. In fact, in any finite dimensional vector space every finite spanning set is a frame. The redundancy of a frame leads to a non-unique representation, however, this makes signal representation resilient to noise and robust to transmission losses. Frames are now standard tools in signal processing and are of great interest to mathematicians and engineers alike. This chapter presents a brief introduction to frames in finite dimensional spaces, and in particular discusses a highly desirable class of frames called equiangular tight frames. Possible research ideas suitable for an undergraduate curriculum are also discussed.

Continuous fields of C*-algebras over finite dimensional spaces

We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as the lower limit (i.e., limit inferior) of the classical derivative of the map where it exists. The new representations lead to significantly shorter proofs for the basic properties of the subgradient and the generalised Jacobian including the chain rule. We establish that a sequence of locally Lipschitz maps between finite dimensional Euclidean spaces converges to a given locally Lipschitz map in the L-topology—that is, the weakest refinement of the sup norm topology on the space of locally Lipschitz maps that makes the generalised Jacobian a continuous functional—if and only if the limit superior of the sequence of directional derivatives of the maps in a given vector direction coincides with the generalised directional derivative of the given map in that direction, with the convergence to the limit superior being uniform for all unit vectors. We then prove our main result that the subspace of Lipschitz C ∞ maps between finite dimensional Euclidean spaces is dense in the space of Lipschitz maps equipped with the L-topology, and, for a given Lipschitz map, we explicitly construct a sequence of Lipschitz C ∞ maps converging to it in the L-topology, allowing global smooth approximation of a Lipschitz map and its differential properties. As an application, we obtain a short proof of the extension of Green’s theorem to interval-valued vector fields. For infinite dimensions, we show that the subgradient of a Lipschitz map on a Banach space is upper continuous, and, for a given real-valued Lipschitz map on a separable Banach space, we construct a sequence of Gateaux differentiable functions that converges to the map in the sup norm topology such that the limit superior of the directional derivatives in any direction coincides with the generalised directional derivative of the Lipschitz map in that direction.

Chapter 5 - Vector Spaces, Hilbert Spaces, and the L2 Space

2. Additions to IV.2. We show below that if S and S) are infinite-dimensional NLS's and if S is a conjugate space (in particular, if S is reflexive), then 53(X, 9)l is not B-convex. This implies that if the conjecture that every B-convex space is reflexive is true, then also the conjecture that every 53(f, J)) is not B-convex for infinite-dimensional S and S) is true (since, by IV.2, if S is not B-convex, then Z ,)) is not B-convex). The principal tool here is the theorem of Aryeh Dvoretzky (see, e.g., Some results on convex bodies and Banach spaces, Proceedings of the International Symposium on Linear Spaces, pp. 123-160, Pergamon Press, New York, 1961) which states that every infinite-dimensional Banach space contains arbitrarily good approximations of finite-dimensional Hilbert space of every finite dimension. Let an(.) be the sequence of period 2n which starts with 2n-I (+ 1)'s and then 2n' (-1)'s. Let k be fixed >2 and let m=2k. Denote the usual basis of 12 by {8n}. Fix E >O and use Dvoretzky's theorem to find xl, ..., xm of unit norm in S such that

Let us remind that by a function we mean a mapping having the domain in the scalar field \( \mathbb{K} \) of real or complex numbers (see Chapter 0, §1). In this chapter, we will consider functions with values in normed spaces — called vector-functions —, as well as functions with values in the field of scalars, which are called scalar-functions. The symbol f(t) means either the function f itself, or the value of f at t; the precise meaning will follow from the context. Prom now on the term “vector-function” will be used in a broader sense, for any mapping \( f:\mathfrak{X} \to \mathfrak{Y} \) with the domain in a finite dimensional space and with values in a normed space.

Application of neustadt's theory of extremals to an optimal control problem with a functional differential equation and a functional inequality constraint

Characterization of simultaneous similarity for commuting m m - tuples of operators is an open problem even in finite-dimensional spaces; known as “A wild problem in linear algebra”. In this paper we offer a criterion for simultaneous similarity of m m -tuples of k k -cyclic commuting operators on an arbitrary Banach space. Moreover, we obtain an additional equivalence condition in the case of finite dimensional Banach spaces, which extends the result found by Shekhtman [Math. Stat. 1 (2013), pp. 157–161] for pairs of cyclic commuting matrices. We also present two applications of our results, one in the case of general multiplication operators on Banach spaces of analytic function, and one for m m -tuples of commuting square matrices.

We shall study in this chapter the existence of fixed points for a different class of mappings, called asymptotically regular mappings. The concept of asymptotically regular mappings is due to Browder and Petryshyn [BP]. Some fixed point theorems for this class of mappings can be found in [Gr1], [Gr2] and references therein. The fixed point theorems which we shall study are based upon results in [DX]. As we shall see, there is a strong connection between these results and those in Chapter VIII. In particular, in some of them the role of the Clarkson modulus of convexity will be played by the moduli of near uniform convexity. In Section 1 we define a new geometric coefficient in Banach spaces which plays the role of the Lifshitz characteristic for asymptotically regular mappings, and we prove the corresponding version for these mappings of Theorem VIII.1.4. In Section 2 we study some relationships between the new coefficient and either the modulus of NUC or the weakly convergent sequence coefficient. We also find a simpler expression for the new coefficient in Banach spaces with the uniform Opial property. Moreover we prove that, in contrast to the Lifshitz characteristic, the new coefficient is easy to compute in lp-spaces. We recall that the Lifshitz characteristic is only known in some renorming of Hilbert spaces.

Separating decompositions of metric spaces are an important randomized clustering paradigm that was formulated by Bartal in [Bar96] and is defined as follows. Given a metric space (X, dx), its modulus of separated decomposability, denoted SEP(X, dx), is the infimum over those σ ∊ (0,∞] such that for every finite subset S ⊆ X and every Δ > 0 there exists a distribution over random partitions P of S into sets of diameter at most Δ such that for every x,y ∊ S the probability that both x and y do not fall into the same cluster of the random partition P is at most σdx (x, y)/Δ. Here we obtain new bounds on SEP(X, ‖·‖x) when (X, ‖· ‖x) is a finite dimensional normed space, yielding, as a special case, that for every n ∊ N. More generally, for every p ∊ [2, ∞]. This improves over the work [CCG+ 98] of Charikar, Chekuri, Goel, Guha, and Plotkin, who obtained this bound when p = 2, yet for p ∊ (2, ∞] they obtained the asymptotically weaker estimate One should note that it was claimed in [CCG+98] that the bound is sharp for every p ∊ [2, ∞], and in particular it was claimed in [CCG+98] that However, the above results show that this claim of [CCG+98] is incorrect for every p ∊ (2, ∞]. Our new bounds on the modulus of separated decomposability rely on extremal results for orthogonal hyperplane projections of convex bodies, specifically using the work [BN02] of Barthe and the author. This yields additional refined estimates, an example of which is that for every n ∊ ℕ and k ∊ {1,…,n} we have where denotesthe subset of ℝn consisting of all those vectors that have at most k nonzero entries, equipped with the Euclidean metric.The above statements have implications to the Lip- schitz extension problem through its connection to random partitions that was developed by Lee and the author in [LN04, LN05]. Given a metric space (X,dx), let e(X) denote the infimum over those K ∊ (0, ∞] such that for every Banach space Y and every subset S ⊆ X, every 1-Lipschitz function f : S → Y has a K-Lipschitz extension to all of X. Johnson, Lindenstrauss and Schechtman proved in [JLS86] that e(X) < dim(X) for every finite dimensional normed space (X, ‖ · ‖x). It is a longstanding open problem to determine the correct asymptotic dependence on dim(X) in this context, with the best known lower bound, due to Johnson and Lindenstrauss [JL84], being that the quantity e(X) must sometimes be at least a constant multiple of In particular, the previously best known upper bound on e(ℓn∞) was the O(n) estimate of [JLS86]. It is shown here that for every n ∊ ℕ we have thus answering (up to logarithmic factors) a question that was posed by Brudnyi and Brudnyi in [BB05, Problem 2]. More generally, for every p ∊ [2, ∞], thus resolving (negatively) a conjecture of Brudnyi and Brudnyi in [BB05, Conjecture 5].