# Seminar on Geometry of Banach Spaces Posts

## Getting continuity of coordinate functionals related to F-basis with logic tools

Given a filter of subsets of natural numbers $$\mathcal{F}$$ we say that a sequence $$(x_n)$$ is $$\mathcal{F}$$ -convergent to $$x$$ if for every $$\varepsilon>0$$ condition $$\{n \in \mathbb N\colon d(x_n , x) < \varepsilon\} \in \mathcal{F}$$ holds. We may use this notion to generalize the idea of Schauder basis, namely we say that a sequence $$(e_n)$$ is an $$\mathcal{F}$$-basis if for every $$x \in X$$ there exists a unique sequence of scalars $$(\alpha_n)$$ s.t. $$\sum_{n,\mathcal{F}} \alpha_n e_n =x$$, which means that the sequence of partial sums is $$\mathcal{F}$$-convergent to $$x$$ Once such a notion is introduced it is natural to ask whenever corresponding coordinate functionals are continuous. Such a question was posed by V. Kadets during the 4th conference Integration, Vector Measures, and Related Topics held in 2011 in Murcia. Surprisingly, there is an obstacle related to the lack of uniform boundedness of functionals related to $$\mathcal{F}$$ basis, due to which we can not find proof of continuity analogous to the classical case. During my talk, I will discuss the problem and provide proof of continuity of considered functionals under some large cardinal assumptions. This is joint work with Tomasz Kania.

## Vector-valued invariant means and projections from the bidual space

Invariant means on amenable groups are an important tool in many parts of mathematics, especially in harmonic analysis. Invariant means and their generalizations for vector-valued functions play also an important role in the stability of functional equations and selections of set-valued functions.

Some generalization of the invariant mean for vector-valued functions was investigated in [2] and the existence of such means is connected with reflexive spaces.

Some generalized definition of an invariant mean has been used by many mathematicians as folklore (e.g. by A. Pełczyński [5]. The explicit form of this definition we can find e.g. in the work of R. Ger [3].

Definition:

Let $$(S,+)$$ be a left [right] amenable semigroup, $$X$$ be a real Banach space. A linear map $$M$$ from the space of all bounded functions from a set $$S$$ into a Banach space $$X$$ into $$X$$ is called left [right] $$X$$–valued invariant mean if
1) norm of $$M$$ is 1;
2) $$M$$ on a constant function is equal this constant,
3) $$M$$ is invariant under a left [right] shifts of the argument of the function.

If $$M$$ is a left and right invariant mean, then $$M$$ is called $$X$$-valued invariant mean.

If in the above definition norm of map $$M$$ is equal at most $$\lambda \geq 1$$, then $$M$$ is called $$X$$-valued invariant $$\lambda$$-mean.

The existence of such invariant means for a fixed Banach space and for all amenable semigroups has been studied by H.B. Domecq [1] (his paper has a gap in the proof which was corrected by T. Kania [4]).

In this talk, we will show a connection between the existence of $X$ -valued invariant $$\lambda$$-means on a Banach $$X$$ and projections from subspaces of bidual space $$X^{**}$$ (with large enough density) onto $$X$$ .

References:
[1] H. Bustos Domecq, Vector-valued invariant means revisited, J. Math. Anal. Appl. 275(2) (2002) 512–520.
[2] Z. Gajda, Invariant means and representations of semigroups in the theory of functional equations, Prace Naukowe Uniwersytetu Śląskiego, Katowice (1992).
[3] R. Ger, The singular case in the stability behavior of linear mappings, Grazer Math. Ber. {316} (1992), 59–70.
[4] T. Kania, Vector-valued invariant means revisited once again, J. Math. Anal. Appl. 445 (2017), 797–802.
[5] A. Pełczyński, Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Rozprawy Mat. 58 (1968), 92 pp.

## O charakteryzacji (i własnościach) $$\Delta$$-przestrzeni $$X$$ (w sensie Reeda) w terminach przestrzeni funkcji ciągłych $$C(X)$$

The seminar will take place in room 1016.

W 1975 roku Reed (zob. [7], [4]) wyróżnił (pod nazwą $$\Delta$$-zbiorów) te nieprzeliczalne podzbiory $$D$$ przestrzeni liczb rzeczywistych $$\mathbb{R}$$ (z topologią naturalną), które mają następujęcą a własność:

Dla każdego malejącego ciągu $$(H_{n})_{n}$$ zbiorów w $$D$$ takich, że $$\bigcap_n H_{_n}=\emptyset$$ istnieje ciąg $$G_{\delta}$$ -zbiorów w $$D$$ taki, że $$H_{n}\subset V_{n},\, n\in\mathbb{N}$$ oraz $$\bigcap_{n}V_{n}=\emptyset.$$

Przymusiński pokazał [6], że istnienie $$\Delta$$ -zbioru w $$\mathbb{R}$$ jest równoważne istnieniu przeliczalnie parazwartej ośrodkowej przestrzeni Moora nie będącej normalną.

Badania wokół $$\Delta$$-zbiorów ściśle związane z badaniami $$\mathbb{Q}$$-zbiorów (w sensie Hausdorffa), te ostatnie nadal stanowią fundamentalne wyzwania w teorii mnogości.

W pracy [2] pojęcie $$\Delta$$ -zbioru zostało rozszerzone do dowolnej przestrzeni topologicznej $$X$$ :

Mówimy, że przestrzeń topologiczna $$X$$ jest $$\Delta$$-przestrzenią a jeżeli dla każdego malejącego ciągu $$(H_{n})_{n}$$ zbiorów w $$X$$ takich, że $$\bigcap_n H_{_n}=\emptyset$$ istnieje ciąg $$(V_{n})_{n}$$ otwartych zbiorów w $$X$$ takich, że $$H_{n}\subset V_{n},\, n\in\mathbb{N}$$, oraz $$\bigcap_{n}V_{n}=\emptyset$$.

Pojęcie to pozwoliło uzyskać charakteryzację tych przestrzeni $$C_{p}(X)$$ (tj. przestrzeni funkcji ciągłych z topologią zbieżności punktowej określonych na przestrzeni Tichonowa $$X$$ ), które są wyróżnione (distingushed spaces). Ta ostatnia własność była intensywnie badana w klasie przestrzeni Frécheta, w szczególności przestrzeni Köthego $$\lambda_{p}(A)$$ .

W pracy [2] pokazuje się, że przestrzeń $$X$$ jest $$\Delta$$ -przestrzenią wtedy i tylko wtedy gdy $$C_{p}(X)$$ jest wyróżniona. Ten analityczny opis pozwolił uzyskać szereg nowych wyników dotyczących $$\Delta$$ -zbiorów i $$\Delta$$-przestrzeni [2], [3], [5]. Alternatywne spojrzenie na wyróżnione przestrzenie $$C_p(X)$$ prezentowane było w [1].

Między innymi pokazano w pracy [2], każda zupełna w senie Čecha $$\Delta$$ -przestrzeń jest rozproszona (scattered) i każda rozproszona zwarta przestrzeń Eberleina jest $$\Delta$$-przestrzenią; istnieją a jednak rozproszone zwarte przestrzenie $$X$$ , które nie są a $$\Delta$$-przestrzeniami, np. $$[0,\omega_{1}]$$ . Każda metryzowalna rozproszona przestrzeń topologiczna jest $$\Delta$$-przestrzenią. Podamy zastosowania tych wyników do badania przestrzeni Banacha $$C(K)$$ oraz przestrzeni $$C_{p}(K)$$ .

[1] J. C. Ferrando, S. A. Saxon, If not distinguished, is $$C_p(X)$$ even close?, Proc. Amer. Math. Soc. 149 (2021) Volume 149, 2583–-2596.

[2] J. Kąkol, A. Leiderman, A characterization of $$X$$ for which spaces $$C_p(X)$$ are distinguished and its applications, Proc Amer. Math. Soc. 8 (2021), 86–99.

[3] J. Kąkol, A. Leiderman, Basic properties of $$X$$ for which spaces $$C_p(X)$$ are distinguished, Proc. Amer. Math. Soc. 8 (2021), 257–280.

[4] R. W. Knight, $$\Delta$$ -Sets, Trans. Amer. Math. Soc. 339 (1993), 45–-60.

[5] A. Leiderman, V. V. Tkachuk, Pseudocompact $\Delta$-spaces are often scattered, Monatshefte für Math. 196 (2021).

[6] T. C. Przymusiński, Normality and separability of Moore spaces, Set-Theoretic Topology, Acad. Press, New York, 1977, 325–-337.

[7] G. M. Reed, On normality and countable paracompactness, Fund. Math. 110 (1980), 145–-152.

## Certain properties of Demyanov difference of convex sets

Adding and subtracting subsets of a vector space plays important role in nonsmooth analysis. There is no obvious difference of sets corresponding to the Minkowski (algebraic, vector) sum of sets. Demyanov difference is strictly related to Clarke subdifferential. We are going to compare Demyanov difference with other possible differences, present its new
properties and pose a few questions.

## Functionals on Lipschitz spaces

We will study continuous linear functionals on Lipschitz spaces with a special focus on those belonging to canonical preduals, the Lipschitz-free spaces. First, we introduce a notion of support applicable to all continuous functionals. Then we will discuss their relation to measures. In particular, we will characterize the functionals represented by measures as those functionals that admit a Jordan-like decomposition into a positive and a negative part. We will see that such decomposition does not exist for all functionals in general, and we will identify the cases when it does.

The talk will be based on joint work with Ramón J. Aliaga.

Link to the talk (access via MS Teams)

## 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.