We use a special tiling for the hyperbolic d-space $\mathbb {H}^d$ for $d=2,3,4$ to construct an (almost) explicit isomorphism between the Lipschitz-free space $\mathcal {F}(\mathbb {H}^d)$ and $\mathcal {F}(P)\oplus \mathcal {F}(\mathcal {N})$, where P is a polytope in $\mathbb {R}^d$ and $\mathcal {N}$ a net in $\mathbb {H}^d$ coming from the tiling. This implies that the spaces $\mathcal {F}(\mathbb {H}^d)$ and $\mathcal {F}(\mathbb {R}^d)\oplus \mathcal {F}(\mathcal {M})$ are isomorphic for every net $\mathcal {M}$ in $\mathbb {H}^d$. In particular, we obtain that, for $d=2,3,4$, $\mathcal {F}(\mathbb {H}^d)$ has a Schauder basis. Moreover, using a similar method, we also give an explicit isomorphism between $\mathrm {Lip}(\mathbb {H}^d)$ and $\mathrm {Lip}(\mathbb {R}^d)$.

Let $X, Y$ be two locally compact Hausdorff spaces and $T:C_0(X)\rightarrow C_0(Y)$ be a standard surjective ɛ-norm-additive map, i.e.

\begin{equation*}\big|\|T(f)+T(g)\|-\|f+g\|\big|\leq \varepsilon,\;{\rm for\;all}\; f, g\in C_0(X).\end{equation*}

Then there exist a homeomorphism $\varphi:Y\rightarrow X$ and a continuous function $\lambda:Y\rightarrow\lbrace\pm1\rbrace$ such that

\begin{equation*}|T(f)(y)-\lambda(y)f(\varphi(y))|\leq\frac{3}{2}\varepsilon,\;{\rm for\;all}\;y\in Y,\;f\in C_0(X).\end{equation*}

The estimate ‘$\frac{3}{2}\varepsilon$’ is optimal. And this result can be regarded as a new nonlinear extension of the Banach–Stone theorem.

We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces, recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most classical separable Banach spaces.

We prove that the infinite-dimensional separable Hilbert space is characterized as the unique separable infinite-dimensional Banach space whose isometry class is closed, and also as the unique separable infinite-dimensional Banach space whose isomorphism class is $F_\sigma $. For $p\in \left [1,2\right )\cup \left (2,\infty \right )$, we show that the isometry classes of $L_p[0,1]$ and $\ell _p$ are $G_\delta $-complete sets and $F_{\sigma \delta }$-complete sets, respectively. Then we show that the isometry class of $c_0$ is an $F_{\sigma \delta }$-complete set.

Additionally, we compute the complexities of many other natural classes of separable Banach spaces; for instance, the class of separable $\mathcal {L}_{p,\lambda +}$-spaces, for $p,\lambda \geq 1$, is shown to be a $G_\delta $-set, the class of superreflexive spaces is shown to be an $F_{\sigma \delta }$-set, and the class of spaces with local $\Pi $-basis structure is shown to be a $\boldsymbol {\Sigma }^0_6$-set. The paper is concluded with many open problems and suggestions for a future research.

Let $H^{\infty}(\Omega,X)$ be the space of bounded analytic functions $f(z)=\sum_{n=0}^{\infty} x_{n}z^{n}$ from a proper simply connected domain Ω containing the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z| \lt 1\}$ into a complex Banach space X with $\left\lVert f\right\rVert_{H^{\infty}(\Omega,X)} \leq 1$. Let $\phi=\{\phi_{n}(r)\}_{n=0}^{\infty}$ with $\phi_{0}(r)\leq 1$ such that $\sum_{n=0}^{\infty} \phi_{n}(r)$ converges locally uniformly with respect to $r \in [0,1)$. For $1\leq p,q \lt \infty$, we denote

\begin{equation*}R_{p,q,\phi}(f,\Omega,X)= \sup \left\{r \geq 0: \left\lVert x_{0}\right\rVert^p \phi_{0}(r) + \left(\sum_{n=1}^{\infty} \left\lVert x_{n}\right\rVert\phi_{n}(r)\right)^q \leq \phi_{0}(r)\right\}\end{equation*}

\begin{equation*}R_{p,q,\phi}(\Omega,X)=\inf \left\{R_{p,q,\phi}(f,\Omega,X): \left\lVert f\right\rVert_{H^{\infty}(\Omega,X)} \leq 1\right\}.\end{equation*}

$R_{p,q,\phi}(\Omega,X)$

$X=\mathcal{B}(\mathcal{H})$

$\mathcal{H}$

We prove some results on weakly almost square Banach spaces and their relatives. On the one hand, we discuss weak almost squareness in the setting of Banach function spaces. More precisely, let $(\Omega,\Sigma)$ be a measurable space, let E be a Banach lattice and let $\nu:\Sigma \to E^+$ be a non-atomic countably additive measure having relatively norm compact range. Then the space $L_1(\nu)$ is weakly almost square. This result applies to some abstract Cesàro function spaces. Similar arguments show that the Lebesgue–Bochner space $L_1(\mu,Y)$ is weakly almost square for any Banach space Y and for any non-atomic finite measure µ. On the other hand, we make some progress on the open question of whether there exists a locally almost square Banach space, which fails the diameter two property. In this line, we prove that if X is any Banach space containing a complemented isomorphic copy of c0, then for every $0 \lt \varepsilon \lt 1$, there exists an equivalent norm $|\cdot|$ on X satisfying the following: (i) every slice of the unit ball $B_{(X,|\cdot|)}$ has diameter 2; (ii) $B_{(X,|\cdot|)}$ contains non-empty relatively weakly open subsets of arbitrarily small diameter and (iii) $(X,|\cdot|)$ is (r, s)-SQ for all $0 \lt r,s \lt \frac{1-\varepsilon}{1+\varepsilon}$.

We give an example of a pair of real Banach spaces such that they are neither linearly isomorphic nor isomorphic with respect to the structure of Birkhoff–James orthogonality, but have mutually homeomorphic geometric structure spaces.

The main purpose of this paper is to study weight-semi-greedy Markushevich bases, and in particular, find conditions under which such bases are weight-almost greedy. In this context, we prove that, for a large class of weights, the two notions are equivalent. We also show that all weight semi-greedy bases are truncation quasi-greedy and weight-superdemocratic. In all of the above cases, we also bring to the context of weights the weak greedy and Chebyshev greedy algorithms—which are frequently studied in the literature on greedy approximation. In the course of our work, a new property arises naturally and its relation with squeeze symmetric and bidemocratic bases is given. In addition, we study some parameters involving the weak thresholding and Chebyshevian greedy algorithms. Finally, we give examples of conditional bases with some of the weighted greedy-type conditions we study.

Suppose that X and Y are two real normed spaces. A map $f:X\rightarrow Y$ is called a min-phase-isometry if it satisfies

$$ \begin{align*} \min\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\min\{\|x+y\|,\|x-y\|\} \quad (x,y\in X). \end{align*} $$

We present properties of min-phase-isometries in the case that Y is strictly convex and show that if a min-phase-isometry f (not necessarily surjective) fixes the origin, then it is phase-equivalent to a linear isometry, that is, $f(x)=\varepsilon (x)g(x)$ for $x\in X$, where $g:X\rightarrow Y$ is a linear isometry and $\varepsilon $ is a map from X to $\{-1,1\}$.

Celebrating 100 years of the Banach contraction principle, we prove some fixed point theorems having all ingredients of the principle, but dealing with common fixed points of a contractive semigroup of nonlinear mappings acting in a modulated topological vector space. This research follows the ideas of the author’s recent papers [‘On modulated topological vector spaces and applications’, Bull. Aust. Math. Soc. 101 (2020), 325–332, and ‘Normal structure in modulated topological vector spaces’, Comment. Math. 60 (2020), 1–11]. Modulated topological vector spaces generalise, among others, Banach spaces and modular function spaces. The interest in modulars reflects the fact that the notions of ‘norm like’ but ‘noneuclidean’ (and not even necessarily convex) constructs to measure a level of proximity between complex objects are frequently used in science and technology. To prove our fixed point results in this setting, we introduce a new concept of Opial sets using analogies with the norm-weak and modular versions of the Opial property. As an example, the results of this work can be applied to spaces like $L^p$ for $p> 0 $, variable Lebesgue spaces $L^{p(\cdot )}$ where $1 \leq p(t) < + \infty $, Orlicz and Musielak–Orlicz spaces.

We construct a nonseparable Banach space $\mathcal {X}$ (actually, of density continuum) such that any uncountable subset $\mathcal {Y}$ of the unit sphere of $\mathcal {X}$ contains uncountably many points distant by less than $1$ (in fact, by less then $1-\varepsilon $ for some $\varepsilon>0$). This solves in the negative the central problem of the search for a nonseparable version of Kottman’s theorem which so far has produced many deep positive results for special classes of Banach spaces and has related the global properties of the spaces to the distances between points of uncountable subsets of the unit sphere. The property of our space is strong enough to imply that it contains neither an uncountable Auerbach system nor an uncountable equilateral set. The space is a strictly convex renorming of the Johnson–Lindenstrauss space induced by an $\mathbb {R}$-embeddable almost disjoint family of subsets of $\mathbb {N}$. We also show that this special feature of the almost disjoint family is essential to obtain the above properties.

Magnitude is a numerical invariant of compact metric spaces, originally inspired by category theory and now known to be related to myriad other geometric quantities. Generalizing earlier results in $\ell _1^n$ and Euclidean space, we prove an upper bound for the magnitude of a convex body in a hypermetric normed space in terms of its Holmes–Thompson intrinsic volumes. As applications of this bound, we give short new proofs of Mahler’s conjecture in the case of a zonoid and Sudakov’s minoration inequality.

We present and thoroughly study natural Polish spaces of separable Banach spaces. These spaces are defined as spaces of norms, respectively pseudonorms, on the countable infinite-dimensional rational vector space. We provide an exhaustive comparison of these spaces with admissible topologies recently introduced by Godefroy and Saint-Raymond and show that Borel complexities differ little with respect to these two topological approaches.

We investigate generic properties in these spaces and compare them with those in admissible topologies, confirming the suspicion of Godefroy and Saint-Raymond that they depend on the choice of the admissible topology.

Any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended in a unique way to a bounded linear operator $\widehat {f} : \mathcal {F}(M) \to \mathcal {F}(N)$ between their corresponding Lipschitz-free spaces. In this paper, we give a necessary and sufficient condition for $\widehat {f}$ to be compact in terms of metric conditions on $f$. This extends a result by A. Jiménez-Vargas and M. Villegas-Vallecillos in the case of non-separable and unbounded metric spaces. After studying the behaviour of weakly convergent sequences made of finitely supported elements in Lipschitz-free spaces, we also deduce that $\widehat {f}$ is compact if and only if it is weakly compact.

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal {F}(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal {F}(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.

We show that every Lorentz sequence space $d(\textbf {w},p)$ admits a 1-complemented subspace Y distinct from $\ell _p$ and containing no isomorph of $d(\textbf {w},p)$ . In the general case, this is only the second nontrivial complemented subspace in $d(\textbf {w},p)$ yet known. We also give an explicit representation of Y in the special case $\textbf {w}=(n^{-\theta })_{n=1}^\infty $ ( $0<\theta <1$ ) as the $\ell _p$ -sum of finite-dimensional copies of $d(\textbf {w},p)$ . As an application, we find a sixth distinct element in the lattice of closed ideals of $\mathcal {L}(d(\textbf {w},p))$ , of which only five were previously known in the general case.

In this article, we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal {L}(E, F)$ . By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set K of $\mathcal {L}(E, F)$ (in the weak operator topology) such that $0$ is an element of its closure (in the weak operator topology) but it is not in its norm-closed convex hull, then we can guarantee the existence of an operator that does not attain its norm. This allows us to provide the following generalisation of results due to Holub and Mujica. If E is a reflexive space, F is an arbitrary Banach space and the pair $(E, F)$ has the (pointwise-)bounded compact approximation property, then the following are equivalent:

1. (i) $\mathcal {K}(E, F) = \mathcal {L}(E, F)$ ;

2. (ii) Every operator from E into F attains its norm;

3. (iii) $(\mathcal {L}(E,F), \tau _c)^* = (\mathcal {L}(E, F), \left \Vert \cdot \right \Vert )^*$ ,

where $\tau _c$ denotes the topology of compact convergence. We conclude the article by presenting a characterisation of the Schur property in terms of norm-attaining operators.

We study super weakly compact operators through a quantitative method. We introduce a semi-norm $\sigma (T)$ of an operator $T:X\to Y$ , where X, Y are Banach spaces, the so-called measure of super weak noncompactness, which measures how far T is from the family of super weakly compact operators. We study the equivalence of the measure $\sigma (T)$ and the super weak essential norm of T. We prove that Y has the super weakly compact approximation property if and and only if these two semi-norms are equivalent. As an application, we construct an example to show that the measures of T and its dual $T^*$ are not always equivalent. In addition we give some sequence spaces as examples of Banach spaces having the super weakly compact approximation property.

We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality

\[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \]

$x,\,y\in X$

$X$

$Y$

$f : X\rightarrow Y$

$\varepsilon : X \rightarrow \{-1,\,1\}$

$\varepsilon \cdot f$

We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality

\[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \]

$x,\,y\in X$

$X$

$Y$

$f : X\rightarrow Y$

$\varepsilon : X \rightarrow \{-1,\,1\}$

$\varepsilon \cdot f$

Let X be a real Banach space. The rectangular constant $\mu (X)$ and some generalisations of it, $\mu _p(X)$ for $p \geq 1$ , were introduced by Gastinel and Joly around half a century ago. In this paper we make precise some characterisations of inner product spaces by using $\mu _p(X)$ , correcting some statements appearing in the literature, and extend to $\mu _p(X)$ some characterisations of uniformly nonsquare spaces, known only for $\mu (X)$ . We also give a characterisation of two-dimensional spaces with hexagonal norms. Finally, we indicate some new upper estimates concerning $\mu (l_p)$ and $\mu _p(l_p)$ .