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Register a new userRevisiting the Hahn-Banach Theorem and Nonlinear Infinite Programming
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[REVISED VERSION] The aim of this paper is to state a sharp version of the Konig supremum theorem, an equivalent reformulation of the Hahn--Banach theorem. We apply it to derive statements of the Lagrange multipliers, Karush-Kuhn-Tucker and Fritz John type, for nonlinear infinite programs. We also show that a weak concept of convexity coming from minimax theory, infsup-convexity, is the adequate one for this kind of results.
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An important consequence of the Hahn-Banach Theorem says that on any locally convex Hausdorff topological space $X$, there are sufficiently many continuous linear functionals to separate points of $X$. In the paper, we establish a `local version of this theorem. The result is applied to study the uo-dual of a Banach lattice that was recently introduced in [3]. We also provide a simplified approach to the measure-free characterization of uniform integrability established in [8].
We study the existence of zeroes of mappings defined in Banach spaces. We obtain, in particular, an extension of the well-known Bolzano-Poincare-Miranda theorem to infinite dimensional Banach spaces. We also establish a result regarding the existence of periodic solutions to differential equations posed in an arbitrary Banach space.
We continue our investigation, from cite{dh}, of the ring-theoretic infiniteness properties of ultrapowers of Banach algebras, studying in this paper the notion of being purely infinite. It is well known that a $C^*$-algebra is purely infinite if and only if any of its ultrapower is. We find examples of Banach algebras, as algebras of operators on Banach spaces, which do have purely infinite ultrapowers. Our main contribution is the construction of a Cuntz-like Banach $*$-algebra which is purely infinite, but does not have purely infinite ultrapowers. Our proof of being purely infinite is combinatorial, but direct, and so differs from the proof for the Cuntz algebra. We use an indirect method (and not directly computing norm estimates) to show that this algebra does not have purely infinite ultrapowers.
Ordered vector spaces E and F are said to be order isomorphic if there is a (not necessarily linear) bijection between them that preserves order. We investigate some situations under which an order isomorphism between two Banach lattices implies the persistence of some linear lattice structure. For instance, it is shown that if a Banach lattice E is order isomorphic to C(K) for some compact Hausdorff space K, then E is (linearly) isomorphic to C(K) as a Banach lattice. Similar results hold for Banach lattices order isomorphic to c_0, and for Banach lattices that contain a closed sublattice order isomorphic to c_0.
Famous Naimark-Han-Larson dilation theorem for frames in Hilbert spaces states that every frame for a separable Hilbert space $mathcal{H}$ is image of a Riesz basis under an orthogonal projection from a separable Hilbert space $mathcal{H}_1$ which contains $mathcal{H}$ isometrically. In this paper, we derive dilation result for p-approximate Schauder frames for separable Banach spaces. Our result contains Naimark-Han-Larson dilation theorem as a particular case.
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