Hahn-Banach Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Separation of Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Generalization of Hahn Banach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Hahn-Banach for C . . . . . . . . . ...

We show spherical completeness of the ring of Colombeau generalized real numbers endowed with the sharp norm. As an application, we establish a Hahn-Banach extension theorem for ultra-pseudo-normed modules of generalized functions in the sense of Colombeau.

This paper discusses under what conditions two disjoint convex subsets of a linear topological space can be separated by a continuous linear functional. The equivalence of several forms of the Hahn-Banach theorem is proven. The separation problem is considered in linear topological spaces, locally convex linear topological spaces, Banach spaces, and finally finite dimensional Banach spaces. A n...

We lift upper and lower estimates from linear functionals to n-homogeneous polynomials and using this result show that l ∞ is finitely represented in the space of n-homogeneous poly-nomials, n ≥ 2, for any infinite dimensional Banach space. Refinements are also given. The classical Dvoretsky spherical sections theorem [5,13] states that l 2 is finitely represented in any infinite dimensional Ba...

Abstract We establish inter alia a compactness criterion in metric spaces involving sequence of completely continuous mappings, which is continuously convergent, the sense H. Hahn, to identity mapping. For Banach spaces, linear version that result coincides with theorem due Mazur. also present new proof Ascoli–Arzelà theorem, we use above applied Bernstein operators.

It is well known that different kinds of expressions of modulus of convexity are essentially based on two geometrical propositions. For one of the propositions we first present a new proof by the Hahn–Banach theorem and intermediate value theorem, then give some corollaries to it which lead to some new expressions of modulus of convexity.