Hahn-Banach Theorem Research Articles

Interpolation type problems in the class of positive trigonometric polynomials of fixed order

This paper is devoted to a family of interpolation type problems for positive trigonometric polynomials of a given ordern. Via the Riesz-Fejer factorization theorem, they can be viewed as natural generalizations of the partial autocorrelation problem for discrete time signals of lengthn+1. The relevant variables for a specific problem are well-defined linear combinations of the coefficients of the underlying trigonometric polynomial. An efficient method is obtained to characterize the feasibility region of the problem, defined as the set of points having these variables as coordinates. It allows us to determine the boundary of that region by computing the extreme eigen values and the corresponding eigenvectors of certain well-defined Hermitian Toeplitz matrices of ordern+1. The method is an extension of one proposed by Steinhardt to solve the coefficient problem for positive cosine polynomials (which belongs to the family). Other interesting applications are the Nevanlinna-Pick interpolation problem for polynomial functions, and the simple interpolation problem for positive trigonometric polynomials. The close connection between the generalized Steinhardt method and classical techniques based on the polarity theorem for convex cones and on the Hahn-Banach extension theorem are established.

A Hahn-Banach theorem for separation of semigroups and its applications

In the first part of this paper a separation theorem is proved for disjoint subsemigroups of a given abelian semigroup. Applying this result, separation theorems and characterization theorems are obtained for semiinternal functions.

Separation theorems for oriented matroids

In this paper we show that Minty's lemma can be used to prove the Hahn-Banach theorem as well as other theorems in this class such as Radon's and Helly's theorem for oriented matroids having an intersection property which guarantees that every pair of flats intersects in some point extension O ∪ p of the oriented matroid O .

On matricially normed spaces

Arveson and Wittstock have proved a non-commutati ve HahnBanach Theorem for completely hounded operator-valued maps on of operators. In this paper it is shown that if T is a linear map from the dual of an operator space into a C*-algebra, then the usual operator norm of T coincides with the completely bounded norm. This is used to prove that the Arveson-Wittstock theorem does not generalize to normed An elementary proof of the Arveson-Wittstock result is presented. Finally a simple bimodule interpretation is given for the Haagerup and matricial tensor products of matricially normed spaces. 1. Introduction. A. function space V on a set X is a linear subspace of the bounded complex functions on X. With the uniform norm, this is a normed vector space. Conversely, any (complex) normed vector space V may be realized as a function space on the closed unit ball X of the dual space V*. Thus one may regard a normed vector space as simply an abstract function space. An operator space V on a Hubert space H is a linear subspace of the bounded operators on H. For each «GN, the operator norm associated with Hn determines a distinguished norm on the n x n matrices over V. The second author recently gave an abstract characterization for the operator by taking into consideration these systems of matrix norms. The operator V are characterized among the normed spaces (see §2), by the L°°-property: given matrices v = [v/7], w = [wkl] with vzy, wkι e V, ||vθH>||=max{||v||,|M|}.

Strong uniqueness of the extension of linear continuous functionals according to the Hahn-Banach theorem

Strong uniqueness of the extension of linear continuous functionals according to the Hahn-Banach theorem

Fundamental properties of linear control systems with after-effect—I: The continuous case

The properties of controllability and output controllability for continuous linear time-varying systems with multiple time-lag are investigated. The obtained results are then used to formulate local parametrical identifiability of such systems using the Hahn–Banach theorem. The use of some feedback control laws guaranteeing the maintenance of the above properties for a class of open-loop controls is also investigated.

Smooth extensions of Lipschitzian real functions

In this short note we point out that any Lipschitzian real function f f defined in a subset K K of a Banach space E E , with span ¯ (K) ≠ E \overline {{\text {span}}} {\text {(K)}} \ne {\text {E}} , can be extended to a surjective, open and Lipschitzian real function g g on E E in such a way that, for every r ∈ R r \in {\mathbf {R}} , the set g − 1 ( r ) {g^{ - 1}}(r) is arcwise connected. In fact, a more refined result is proved.

Multilinear maps and tensor norms on operator systems

We extend work of Christensen and Sinclair on completely bounded multilinear forms to the case of subspaces of C ∗ algebras, and obtain a representation theorem and a Hahn-Banach extension theorem for such maps. In the second part of the paper the Haagerup norms on tensor products are investigated, and we obtain new characterizations of these quantities.

Infinite-dimensional Gale-Nikaido-Debreu theorem and a fixed-point theorem of Tarafdar

This paper proves the equivalence of five fixed-point theorems in topological vector spaces and then uses them to deduce an infinite-dimensional generalization of the classic Gale-Nikaido-Debreu theorem. The proof is based on the Hahn-Banach theorem and the Tarafdar fixed point theorem. There is also a discussion of the relationship of this generalization to the existence theorems of Aliprantis and Brown, Border, Yannelis and others.

On the abstract Hahn-Banach theorem due to Rodé

On the abstract Hahn-Banach theorem due to Rodé

Another ‘short’ proof of the Riesz representation theorem

1. Introduction. In [2], a short synthetic proof of the Riesz representation theorem was given; this used the Hahn-Banach theorem, the Stone-Čech compactification of a discrete space and the Caratheodory extension procedure for measures. In this note, we show how the theorem can be proved using ultrapowers in place of the Stone-Čech compactification. We also describe how the proof can be expressed in a non-standard way (a rather different non-standard proof has been given by Loeb [4]).

The Kreĭn-Šmulian theorem

PROOF. Let {xi} be a dense subset of X. Suppose that d(x**, JX) = d > 0. By the Hahn-Banach theorem there is a norm one functional x*** satisfying x***(JX) =0 and x***(x**)= d. Let Wn = {z*: Iz*(xi)l d/2} n W,. The sequence {x*} converges to zero in the weak* topology, for given x and E > 0, there is an xj with II(x/E) xjll j; yet Ix**(x *)I > d/2 for all n. Q.E.D.

Which set existence axioms are needed to prove the separable Hahn-Banach theorem?

We work in the context of weak subsystems of second order arithmetic. RCA 0 is the system with Δ 1 0 comprehension and Σ 1 0 induction on the natural numbers. WKL 0 is RCA 0 plus weak König's lemma for trees of finite sequences of 0's and 1's. Within RCA 0 we encode a separable Banach space  as a countable normed space A over Q . Points of  are Cauchy sequences from A which converge at the rate of at least 2 − n . We show that the Hahn-Banach theorem for separable Banach spaces is provably equivalent to WKL 0 over RCA 0. Thus, once again, WKL 0 is revealed as mathematically powerful, despite being proof theoretically equivalent to primitive recursive arithmetic.

VECTOR SPACES WITH STRICTLY *-EQUABLE CONVERGENCE STRUCTURES

ABSTRACT Certain results in connection with boundedness and precompactness, which are valid in topological vector spaces and locally bounded prelimit vector spaces, are extended to the class of strictly *-equable pre-limit vector spaces. This class contains the class of topological vector spaces. Hahn-Banach extension and separation theorems hold in the class of strictly *-equable prelimit vector spaces.

On means with countably additive continuities

Some notions of countable additivity, meaningful for an expectation whose domain is an arbitrary linear space of bounded, real-valued functions, are studied. For linear spaces or ordered linear spaces which do not possess the structure of a vector lattice important in the development of the Lebesgue-Daniell integralthere is to our knowledge no study of continuous or countably additive linear functionals other than the seminal contributions of de Finetti [1 and 2]. The present paper was inspired mainly by de Finetti's writings. For simplicity, linear subspaces L of lo (Q), the space of bounded R-valued (realvalued) functions defined on a nonempty set Q, will be considered and, on L, only linear functions Q, necessarily nonnegative, which satisfy f C L and a < f < b everywhere imply a < Qf < b. Call such a functional a prevision, a term borrowed from de Finetti [1 and 2]. With the help of the Hahn-Banach extension theorem, it is simple to verify that each prevision on L is the restriction to L of a prevision P on lo. Let Ap be the collection of all L such that P restricted to L is continuous or countably additive. In [1, Chapter 5.34 and 2, Vol. 2, Appendix 18.3], de Finetti introduces Ap and initiates the study of its structure. He observes L' c L C Ap implies L' C Ap, and he goes on to state L1 and L2 belong (to Ap), then so does L1 + L2 (the linear space of sums X1 + X2, X1 E L1 and X2 c L2). The present paper developed in large part from the observation that the quoted assertion is erroneous. Several counterexamples, which also illustrate other phenomena, are offered. For the first, [0,1] designates the closed unit interval, C = C[0,11] the space of all continuous R-valued functions with [0,1] as their domain, and A is the usual Lebesgue integral. The indicator of a set is, as usual, the function that assumes the value 1 on the set 0 off the set. The useful convention introduced by de Finetti of designating a set and its indicator by the same letter will usually be adopted here. EXAMPLE 1. Let Wp be an open dense subset of [0,1] with Ap < 1, L1 = C[0, 1], L2 = {t: t C R}, L = L1 + L2 and P(O + t) = AO + t, 0 e C[0, 1]. To formulate the strong sense in which Example 1 is a counterexample, it is necessary to articulate two definitions. If Pfn -? 0 for every decreasing sequence fC e L that converges pointwise to 0, call P m-continuous on L (m for 'monotone'). Let v(L) designate the smallest sigma field of subsets of Q such that f-1 (B) C a(L) Received by the editors February 21, 1983 and, in revised form, August 29, 1983. 1980 Mathematics Subject Cossification. Primary 28C05, 46G12, 60A05.

B*-algebra representations in a quaternionic Hilbert module

It is shown that the Gel’fand–Naimark–Segal (GNS) construction can be generalized to real B*-algebras containing an algebra *-isomorphic to the quaternion algebra by the use of quaternion linear functionals and Hilbert Q-modules. An extension of the Hahn–Banach theorem to such functionals is proved.

Thermodynamics based on the Hahn-Banach Theorem: The Clausius inequality

Thermodynamics based on the Hahn-Banach Theorem: The Clausius inequality

C∞ functions in infinite dimension and linear partial differential difference equations with constant coefficients

This work is closed to [2] where a dense linear subspace \(\mathbb{E}\)(E) of the space ℰ(E) of the Silva C ∞ functions on E is defined; the dual of \(\mathbb{E}\)(E) is described via the Fourier transform by a Paley-Wiener-Schwartz theorem which is formulated exactly in the same way as in the finite dimensional case. Here we prove existence and approximation result for solutions of linear partial differential difference equations in \(\mathbb{E}\)(E) with constant coefficients. We also obtain a Hahn-Banach type extension theorem for some C∞ functions defined on a closed subspace of a DFN space, which is analogous to a Boland’s result in the holomorphic case [1].

Calculus on complex Banach spaces

By using the techniques of modern functional analysis, a variety of new concepts have been developed and new results proved which extend considerably the new calculus on complex Banach spaces developed by Sharma and Rebelo. The distinguishing feature of the new calculus is that in this calculus the more general concept of additivity replaces that of linearity in the Frechet calculus. It is proved that the space of continuous additive maps between two complex Banach spaces is the direct sum of the spaces of linear and semilinear maps between the two spaces. The Hahn-Banach theorem and the open mapping theorem which in their standard versions are valid for continuous linear functionals and functions are shown to hold also for the additive case. The concepts of the adjoint of an additive map, of a new kind of orthogonal complement of a subset of a Banach space, and of a balanced additive map in which the norms of the linear and semilinear components are equal are developed. It is then proved that the orthogonal complement of the range of an additive map equals the null space of its adjoint and if the additive map is a functional on a complex Hilbert space and is balanced, then the orthogonal complement of the null space of the functional equals the range of the adjoint. A generalization of the inverse function theorem is proved by using our version of the open mapping theorem and then used to establish the Lagrange multiplier theorem in the new calculus. A number of related results are also proved. The applications of the new calculus to physics are briefly described.

A generalization of the Hahn-Banach theorem

A generalization of the Hahn-Banach theorem