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**The Hahn-Banach Theorem: a proof of the equivalence between the analitic and geometric versions(****arXiv****)**

**Author :** Fidel José Fernández y Fernández Arroyo

**Abstract : **We present here a simple and direct proof of the classic geometric version of Hahn-Banach Theorem from its analitic version, in the real case. The reciprocal implication, and the direct proofs of both versions, are already well kown, but they are also summarized. For the complex case, in both versions the Hahn-Banach Theorem is deduced from the real case, as it is well known.

**2. A Local Hahn-Banach Theorem and Its Applications(****arXiv)**

**Author : **Niushan Gao, Denny H. Leung, Foivos Xanthos

**Abstract : **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]

**3.On metric convexity, the discrete Hahn-Banach theorem, separating systems and sets of points forming only acute angles (****arXiv****)**

**Author : **Hugues Randriambololona

**Abstract : **This text has three parts. The first one is largely autobiographical, hence my use of the first person. There I recall how Gerard Cohen influenced important parts of my research. The second is of a more classic mathematical nature. I present a discrete analogue of the Hahn-Banach theorem, which serves as a basis for generalizing the notion of separating systems in the context of metric convexity. The third one aims at building a bridge between two communities of researchers, those interested in separating systems, and those interested in a certain question in combinatorial geometry — — sets of points forming only acute angles — — who seem not to be aware of each other, while they are working on precisely the same problem! Of course, these three themes are closely intertwined.

**4. The Tracial Hahn-Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra(****arXiv****)**

**Author :** J. William Helton, Igor Klep, Scott McCullough

**Abstract : **This article investigates matrix convex sets and introduces their tracial analogs which we call contractively tracial convex sets. In both contexts completely positive (cp) maps play a central role: unital cp maps in the case of matrix convex sets and trace preserving cp (CPTP) maps in the case of contractively tracial convex sets. CPTP maps, also known as quantum channels, are fundamental objects in quantum information theory. Free convexity is intimately connected with Linear Matrix Inequalities (LMIs) L(x) = A_0 + A_1 x_1 + … + A_g x_g > 0 and their matrix convex solution sets { X : L(X) is positive semidefinite }, called free spectrahedra. The Effros-Winkler Hahn-Banach Separation Theorem for matrix convex sets states that matrix convex sets are solution sets of LMIs with operator coefficients. Motivated in part by cp interpolation problems, we develop the foundations of convex analysis and duality in the tracial setting, including tracial analogs of the Effros-Winkler Theorem. The projection of a free spectrahedron in g+h variables to g variables is a matrix convex set called a free spectrahedrop. As a class, free spectrahedrops are more general than free spectrahedra, but at the same time more tractable than general matrix convex sets. Moreover, many matrix convex sets can be approximated from above by free spectrahedrops. Here a number of fundamental results for spectrahedrops and their polar duals are established. For example, the free polar dual of a free spectrahedrop is again a free spectrahedrop. We also give a Positivstellensatz for free polynomials that are positive on a free spectrahedrop.

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