The classical Hahn-Banach Theorem states that any linear bounded functional defined on a linear subspace of a normed space admits a norm-preserving linear bounded extension to the whole space. The constructive and computational content of this theorem has been studied by Bishop, Bridges, Metakides, Nerode, Shore, Kalantari, Downey, Ishihara and others and it is known that the theorem does not a...

Anthony J. Hynes 1,*, Cheryl Tatum Ernest 1,†, Deanna Donohoue 1,‡, Dieter Bauer 1 and Arnout Ter Schure 2 1 Division of Marine and Atmospheric Chemistry, Rosenstiel School of Marine and Atmospheric Science, University of Miami, 4600 Rickenbacker Causeway, Miami, FL 33149, USA; [email protected] (C.T.E.); [email protected] (D.D.); [email protected] (D.B.) 2 Electric Power ...

We consider orthogonal polynomials in two variables whose derivatives with respect to x are orthogonal. We show that they satisfy a system of partial differential equations of the form α(x, y)∂ x −→ Un + β(x, y)∂x −→ Un = Λn −→ Un, where degα ≤ 2, deg β ≤ 1, −→U n = (Un0, Un−1,1, · · · , U0n) is a vector of polynomials in x and y for n ≥ 0, and Λn is an eigenvalue matrix of order (n+ 1)× (n+ 1)...

Let T̃ = ∑n i=1 ũi⊗ vi : V → V = [v1, ..., vn] ⊂ X, where ũi ∈ V ∗ and X is a Banach space. Let T = ∑n i=1 ui ⊗ vi : X → V be an extension of T̃ to all of X (i.e., ui ∈ X∗) such that T has minimal (operator) norm. In this paper we show in particular that, in the case n = 2 and the field is R, there exists a rank-n T̃ such that ‖T‖ = ‖T̃‖ for all X if and only if the unit ball of V is either not smo...