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# Seminar on Geometry of Banach Spaces Posts

## On a ρ-orthogonality

ρ-orthogonality is one among many possible definitions of orthogonality in general normed spaces, related to the “derivatives” of the norm.

The talk is based on the article (of the same title) by Jacek Chmieliński and Paweł Wójcik (Aequat. Math. 2010). Apart from basic properties of the introduced notion, the class of maping preserving ρ-orthogonality will be considered.

## The complexity of isometry classes of Banach spaces

I will present our joint work with M. Doucha, M. Doležal and O. Kurka, where we develop a new natural topological approach to coding of separable Banach spaces. It makes meaningful questions such as which Banach spaces are the easiest to define, up to isometry (and also up to isomorphism), or which classes of Banach spaces are the easiest to define – in the descriptive set-theoretic framework. The plan is first to motivate our choice of the topological framework (first half of the seminar) and then (in the second half of the seminar) to concentrate on our results concerning complexities: among the results, there are new characterizations of the separable infinite-dimensional Hilbert space as the separable infinite-dimensional Banach space whose both isometry and isomorphism classes are the easiest to define among Banach spaces (such statements are made absolutely precise); and precise characterization of the complexity of the isometry classes of the most classical Banach spaces such as $$L_p[0,1]$$, $$\ell_p$$, for finite $$1\leq p$$, $$c_0$$, and also of the Gurarii space. The paper opens a new area for research and at the end, I will suggest several open problems.

## Equilateral sets in large Banach spaces

A subset of a Banach space is called equilateral if the distances between any two of its distinct points are the same. In this talk it will be shown that there exist non separable Banach spaces with no uncountable equilateral sets and indeed non separable Banach spaces with no infinite equilateral sets.

Link to the talk.

## On the Schreier space

Link to the talk.

## Any isometry between the spheres of 2-dimensional Banach spaces is linear

Abstract: The famous Tingley’s Problem asks whether every onto isometry between the unit spheres of Banach spaces can be extended to an isometry between the spaces – in this case, the Mazur–Ulam Theorem ensures that the isometry between the spaces would be linear. This Problem is far from been solved, but at least the two-dimensional case has been solved recently in a series of works by the author and Professor Tarás Banakh. In this seminar we will show how a miscellany of several kinds of tools (single variable calculus, linear algebra, differential equations) has allowed to prove the following statement: If X and are two-dimensional Banach spaces, then every isometry between their unit spheres extends to an isometry between X and Y.

### Lorentz spaces and non-differentiability of functions

Abstract: A famous theorem by Rademacher states that Lipschitz functions are differentiable almost everywhere. In this talk, we will look at Sobolev spaces with derivatives in Lorentz spaces and an infinitesimal Lipschitz constant to investigate to which extent Rademacher’s theorem can be generalized.

## Lipschitz geometry of operator spaces and Lipschitz-free operator spaces

Abstract: While the nonlinear geometry of Banach spaces has been extensively studied (especially in the past few decades), the nonlinear geometry of its noncommutative counterpart, i.e., of operator spaces, has been neglected until very recently. In this talk, I will discuss some recent developments in this field. In particular, I will introduce the notion of almost complete Lipschitz embeddability between operator spaces and explain why this leads to a nontrivial nonlinear theory. For that, Lipschitz free spaces of operator spaces will play an important role. (This is a joint with with Javier Alejandro Chávez-Domínguez and Thomas Sinclair).

## When are maps preserving semi-inner products linear?

We observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth Banach space and a mapping that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediateextension of the former result to infinite dimensions, even under an extra hypothesis of uniform smoothness. Regrettably, this result refutes a claim concerning smooth spaces from a recent paper [Aequationes Math. (2020)].