Domingo García

This work is a thorough and detailed study on the geometry of the unit sphere of certain Banach spaces of homogeneous polynomials in ${\mathbb{R}}^2$. Specifically, we provide a complete description of the unit spheres, identify the extreme points of the unit balls, derive explicit formulas for the corresponding polynomial norms, and describe the techniques required to tackle these questions. To enhance the comprehensiveness of this work, we complement the results and their proofs with suitable diagrams and figures. The new results presented here settle some open questions posed in the past. For the sake of completeness of this work, we briefly discuss previous known results and provide directions of research and applications of our results.

We study the set $${\text {MA}}(X,Y)$$MA ( X , Y ) of operators between Banach spaces X and Y that attain their minimum norm, and the set $${\text {QMA}}(X,Y)$$QMA ( X , Y ) of operators that quasi attain their minimum norm. We characterize the Radon–Nikodym property in terms of operators that attain their minimum norm and obtain some related results about the density of the sets $${\text {MA}}(X,Y)$$MA ( X , Y ) and $${\text {QMA}}(X,Y)$$QMA ( X , Y ) . We show that every infinite-dimensional Banach space X has an isomorphic space Y, such that not every operator from X to Y quasi attains its minimum norm. We introduce and study Bishop–Phelps–Bollobás type properties for the minimum norm, including the ones already considered in the literature, and we exhibit a wide variety of results and examples, as well as exploring the relations between them.

We study the set $\operatorname{MA}(X,Y)$ of operators between Banach spaces $X$ and $Y$ that attain their minimum norm, and the set $\operatorname{QMA}(X,Y)$ of operators that quasi attain their minimum norm. We characterize the Radon-Nikodym property in terms of operators that attain their minimum norm and obtain some related results about the density of the sets $\operatorname{MA}(X,Y)$ and $\operatorname{QMA}(X,Y)$. We show that every infinite-dimensional Banach space $X$ has an isomorphic space $Y$ such that not every operator from $X$ to $Y$ quasi attains its minimum norm. We introduce and study Bishop-Phelps-Bollobás type properties for the minimum norm, including the ones already considered in the literature, and we exhibit a wide variety of results and examples, as well as exploring the relations between them.

We prove the following Farkas’ Lemma for simultaneously diagonalizable bilinear forms: If $$A_1,\ldots ,A_k$$A 1 , … , A k , and $$B:\mathbb {R}^n \times \mathbb {R}^n \rightarrow \mathbb {R}$$B : R n × R n → R are bilinear forms, then one—and only one—of the following holds: $$B=a_1 A_1 + \cdots + a_k A_k,$$B = a 1 A 1 + ⋯ + a k A k , with non-negative $$a_i\text {'s}$$a i 's , there exists (x, y) for which $$A_1(x,y) \ge 0 , \ldots , A_k(x,y) \ge 0$$A 1 ( x , y ) ≥ 0 , … , A k ( x , y ) ≥ 0 and $$B(x,y) < 0$$B ( x , y ) < 0 . We study evaluation maps over the space of bilinear forms and consequently construct examples in which Farkas’ Lemma fails in the bilinear setting.