# Seminar on Geometry of Banach Spaces Posts

## 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)].