prof. Mgr. Petr Hájek, DrSc.

In the present paper we introduce and study the Lipschitz retractional structure of metric spaces. This topic was motivated by the analogous projectional structure of Banach spaces, a topic that has been thoroughly investigated. The more general metric setting fits well with the currently active theory of Lipschitz free spaces and spaces of Lipschitz functions. Among our applications we show that the Lipschitz free space F (X) is a Plichko space whenever X is a Plichko Banach space. Our main results include two examples of metric spaces. The first one M contains two points {0, 1} such that no separable subset of M containing these points is a Lipschitz retract of M . The second example fails the analogous property for arbitrary infinite density. Finally, we introduce the metric version of the concept of locally complemented Banach subspace, and prove some metric analogues to the linear theory.

The Banach space P(X-2) of 2-homogeneous polynomials on the Banach space X can be naturally embedded in the Banach space Lip(0) (B-X) of real-valued Lipschitz functions on Bx that vanish at 0. We investigate whether P(X-2) is a complemented subspace of Lip(0) (B-X). This line of research can be considered as a polynomial counterpart to a classical result by Joram Lindenstrauss, asserting that P(X-1) = X* is complemented in Lip(0) (B-X) for every Banach space X. Our main result asserts that P(X-2) is not complemented in Lip(0) (B-X) for every Banach space X with non-trivial type.

In the present note we give two explicit constructions (based on a retractional argument) of a Schauder basis for the Lipschitz free space F(N), over certain uniformly discrete metric spaces N. The first one applies to every net N in a finite dimensional Banach space, leading to the basis constant independent of the dimension. The second one applies to grids in Banach spaces with an FDD. As a corollary, we obtain a retractional Schauder basis for the Lipschitz free space F(N) over a net N in every Banach space X with a Schauder basis containing a copy of c(0), as well as in every Banach space with a c(0)-like FDD. (C) 2022 Elsevier Inc. All rights reserved.

We prove that a 2n-dimensional real normed space is Hilbertian (i.e. Euclidean) if it admits two distinct decompositions into an l(2)-sum of two n-dimensional subspaces. We give an approximate version of this result as well as an infinite-dimensional version of the result. Finally, we note that the complex version of the result also holds. (C) 2020 Elsevier Inc. All rights reserved.

We construct an Asplund Banach space X with a norming Markusevic basis such that X is not weakly compactly generated. This solves a long-standing open problem from the early nineties, originally due to Gilles Godefroy. En route to the proof, we construct a peculiar example of scattered compact space, that also solves a question due to Wieslaw Kubis and Arkady Leiderman. (c) 2021 Elsevier Inc. All rights reserved.

We consider the following problem (which is a generalisation of a folklore result Proposition 1 below): given a continuous linear operator T : X -> Y, where Y is a Banach space with a (long) sub-symmetric basis, under which conditions can we find a continuous linear operator S : X -> Y such that S(B-X) contains the basis of Y. As a tool we also consider a non-separable version of Theorem 2 below: Given an infinite subset A subset of X*, under which conditions can we find a biorthogonal system in X x A of cardinality card A?

We show that finite dimensional Banach spaces fail to be uniformly non locally almost square. Moreover, we construct an equivalent almost square bidual norm on l(infinity). As a consequence we get that every dual Banach space containing c(0) has an equivalent almost square dual norm. Finally we characterize separable real almost square spaces in terms of their position in their fourth duals. (C) 2020 The Authors. Published by Elsevier Inc.

In the first part of our note we prove that every Weakly Lindelof Determined (WLD) (in particular, every reflexive) non-separable Banach X space contains two dense linear subspaces Y and Z that are not densely isomorphic. This means that there are no further dense linear subspaces Y-0 and Z(0) of Y and Z which are linearly isomorphic.

The paper elucidates the relationship between the density of a Banach space and possible sizes of Auerbach systems and well-separated subsets of its unit sphere. For example, it is proved that for a large enough space X, the unit sphere SX always contains an uncountable (1+)-separated subset. In order to achieve this, new results concerning the existence of large Auerbach systems are established, that happen to be sharp for the class of weakly Lindelof determined (WLD) spaces. In fact, we offer the first consistent example of a non-separable WLD Banach space that contains no uncountable Auerbach system, as witnessed by a renorming of c(0)(omega(1)). Moreover, the following optimal results for the classes of, respectively, reflexive and super-reflexive spaces are established: the unit sphere of an infinite-dimensional reflexive space contains a symmetrically (1 + epsilon)-separated subset of any regular cardinality not exceeding the density of X; should the space X be super-reflexive, the unit sphere of X contains such a subset of cardinality equal to the density of X. The said problem is studied for other classes of spaces too, including WLD spaces, RNP spaces, or strictly convex ones.

In the first part of our paper, we show that too has a dense linear subspace which admits an equivalent real analytic norm. As a corollary, every separable Banach space, as well as ti (c), also has a dense linear subspace which admits an analytic renorming. By contrast, no dense subspace of ce(Loi) admits an analytic norm. In the second part, we prove (solving in particular an open problem of Guirao, Montesinos, and Zizler in [7]) that every Banach space with a long unconditional Schauder basis contains a dense subspace that admits a C -smooth norm. Finally, we prove that there is a proper dense subspace of 2,c,o(w1) that admits no Gateaux smooth norm. (Here, Ego (cal) denotes the Banach space of real-valued, bounded, and countably supported functions on col.) (C) 2020 Elsevier Inc. All rights reserved.

In this note we are concerned with the validity of an uncountable analogue of a combinatorial lemma due to Vlastimil Ptak. We show that the validity of the result for col can not be decided in ZFC alone. We also provide a sufficient condition, for a class of larger cardinals. (C) 2018 Elsevier Inc. All rights reserved.

We generalize a result on Banach spaces with separable dual which was first shown by Zippin, and was explicitly formulated by Benyamini. We prove that there is a class of Asplund spaces, which includes all spaces with separable dual, whose members can be perturbed inside a suitable ambient space to be contained in the space of continuous functions on a well-founded compact tree.

We construct a zero dimensional uniform Eberlein compact space K such that C(K) is a Hilbert generated Banach space which does not admit any norming Markushevich basis. This solves a classical problem from the seventies. (C) 2019 Elsevier Inc. All rights reserved.

Let X be a non-separable super-reflexive Banach space. Then for any separable Banach space Y of dimension at least two there exists a C-infinity-smooth surjective mapping f : X -> Y such that the restriction of f onto any separable subspace of X fails to be surjective. This solves a problem posed by Aron, Jaramillo, and Ransford (Problem 186 in the book [5]). (C) 2017 Elsevier Inc. All rights reserved.

Let Gamma be an uncountable cardinal. We construct a real symmetric 1-Lipschitz function on the unit sphere of c(0)(Gamma) whose restriction to any nonseparable subspace is a distortion.

Let lambda be a large enough cardinal number (assuming the Generalized Continuum Hypothesis it suffices to let lambda = N-omega). If X is a Banach space with dens(X) >= lambda, which admits a coarse (or uniform) embedding into any c(0)(Gamma), then X fails to have non-trivial cotype, i.e. X contains l(infinity)(n) C-uniformly for every C > 1. In the special case when X has a symmetric basis, we may even conclude that it is linearly isomorphic with c(0) (dens X).

We show that if x is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at x, then x is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the sufficient conditions mentioned.

We prove the symmetric version of Kottman's theorem, that is to say, we demonstrate that the unit sphere of an infinite dimensional Banach space contains an infinite subset A with the property that ||x +/- y|| > 1 for distinct elements x, y is an element of A, thereby answering a question of J.M.F. Castillo. In the case where X contains an infinite-dimensional separable dual space or an unconditional basic sequence, the set A may be chosen in a way that ||x +/- y|| >= 1 + epsilon for some epsilon > 0 and distinct x, y is an element of A. Under additional structural properties of X, such as non-trivial cotype, we obtain quantitative estimates for the said epsilon. Certain renorming results are also presented. (C) 2018 Elsevier Inc. All rights reserved.

We prove that in every separable Banach space X with a Schauder basis and a C-k-smooth norm it is possible to approximate, uniformly on bounded sets, every equivalent norm with a C-k-smooth one in a way that the approximation is improving as fast as we wish on the elements depending only on the tail of the Schauder basis. Our result solves a problem from the recent monograph of Guirao, Montesinos and Zizler. (C) 2017 Published by Elsevier Inc.

We give several structural results concerning the Lipschitz-free spaces F(M), where M is a metric space. We show that F(M) contains a complemented copy of l(1)(Gamma), where Gamma = dens(M). If N is a net in a finite dimensional Banach space X, we show that F(N) is isomorphic to its square. If X contains a complemented copy of l(p), c(0) then F(N) is isomorphic to its l(r)-sum. Finally, we prove that for all X congruent to C(K) spaces, where K is a metrizable compact, F(N) are mutually isomorphic spaces with a Schauder basis.

We prove that for any separable Banach space X, there exists a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space contains a complemented subspace isomorphic to X. As a consequence we give an example of a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space fails the approximation property and we prove that there exists an uncountable family of topologically equivalent distances on the Cantor space whose free spaces are pairwise non isomorphic. We also prove that the free space over a countable compact metric space has the Schur property. These results answer questions by G. Godefroy.

We prove that for any separable Banach space X, there exists a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space contains a complemented subspace isomorphic to X. As a consequence we give an example of a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space fails the approximation property and we prove that there exists an uncountable family of topologically equivalent distances on the Cantor space whose free spaces are pairwise non isomorphic. We also prove that the free space over a countable compact metric space has the Schur property. These results answer questions by G. Godefroy.

Let X be a separable infinite dimensional real Banach space. We denote by F(X)F(X) the class of continuous functions f:R×X→Xf:R×X→X such that the ODE View the MathML sourceu′=f(t,u),u(t0)=x,t0∈R,x∈X, has a global solution for any initial condition. Our main result states that A⊂XA⊂X is the cross-section of a solution funnel of the ODE u′=f(t,u),u(0)=0u′=f(t,u),u(0)=0, for some f∈F(X)f∈F(X), if and only if A is an analytic set.

We present an equivalent midpoint locally uniformly rotund (MLUR) renorming of C[0, 1] with the diameter 2 property (D2P), i.e. every non-empty relatively weakly open subset of the unit ball has diameter 2. An example of an MLUR space with the D2P and with convex combinations of slices of arbitrarily small diameter is also given.

In 1970 Haskell Rosenthal proved that if X is a Banach space, I' is an infinite index set, and T : l(infinity)(Gamma) -- X is a bounded linear operator such that inf-yEr 11T(67)11 > 0 then T acts as an isomorphism on l(infinity)(Gamma'), for some r, C P of the same cardinality as P. Our main result is a nonlinear strengthening of this theorem. More precisely, under the assumption of GCH and the regularity of I', we show that if F : Bl(infinity)(r) X is uniformly differentiable and such that infer 11F(e-y) F(0)11 > 0 then there exists x E l(infinity)(Gamma) such that dF(x)[.] is a bounded linear operator which acts as an isomorphism on l(infinity)(Gamma), for some Gamma' C r of the same cardinality as r. (C) 2016 Published by Elsevier Inc.

In 1970 Haskell Rosenthal proved that if X is a Banach space, Γ is an infinite index set, and T:ℓ∞(Γ) → X is a bounded linear operator such that infγ∈Γ ||T(eγ)|| > 0 then T acts as an isomorphism on ℓ∞(Γ'), for some Γ' ⊂Γ of the same cardinality as Γ. Our main result is a nonlinear strengthening of this theorem. More precisely, under the assumption of GCH and the regularity of Γ, we show that if F:Bℓ∞(Γ) → X is uniformly differentiable and such that infγ∈Γ ||F(eγ)-F(0)|| > 0 then there exists x ∈ Bℓ ∞(Γ) such that dF(x)[·] is a bounded linear operator which acts as an isomorphism on ℓ∞(Γ'), for some Γ' ⊂Γ of the same cardinality as Γ.

We give a corrected proof of the main Lemma 2 from the paper in the title (our Corollary 7).

Let X be a WCG Banach space admitting a Ck -smooth norm where k ε ℕ ∪ {∞}. Then X admits an equivalent norm which is simultaneously, C1 -smooth, LUR, and the limit of a sequence of Ck -smooth norms. If X = C([0,α]), where α is any ordinal, then the same conclusion holds true with k = ∞. © 2013 by the authors 1973.

A typical result in this note is that if X is a Banach space which is a weak Asplund space and has the omega*-omega- Kadets Klee property, then X is already an Asplund space.

The main result implies that the Lipschitz-free spaces F(l(1)) and F(R-n) have a Schauder basis. This improves (in a special case) on the previous work of Godefroy and Kalton who showed that F(X) has a bounded approximation property if and only if the Banach space X does. (C) 2014 Elsevier Inc. All rights reserved.

The main result implies that the Lipschitz-free spaces F(ℓ1) and F(Rn) have a Schauder basis. This improves (in a special case) on the previous work of Godefroy and Kalton who showed that F(X) has a bounded approximation property if and only if the Banach space X does.

Let A(n)(X) be the algebra of polynomials on a real Banach space X, which is generated by all continuous polynomials of degree not exceeding n. Let m be the minimal integer such that there is a non-compact m-homogeneous polynomial P epsilon P((m) X; l(1)). Then n >= m implies that the uniform closure of A(n)(X) does not contain all polynomials of degree n + 1, and hence the chain of closures <(A(n)(X))over bar>, n >= m is strictly increasing. In the rest of the note we give solutions to three problems concerning the behaviour of smooth functions on Banach spaces posed in the literature. In particular, we construct an example of a uniformly differentiable real valued function f on the unit ball of a certain Banach space X, such that there exists no uniformly differentiable function g on lambda B-X, for any lambda > 1, which coincides with f in some neighbourhood of the origin. (C) 2013 Elsevier Inc. All rights reserved.

This bookis aboutthe subject of higher smoothness in separable real Banach spaces.It brings together several angles of view on polynomials, both in finite and infinite setting.Also a rather thorough and systematic view of the more recent results, and the authors work is given. The book revolves around two main broad questions: What is the best smoothness of a given Banach space, and its structural consequences? How large is a supply of smooth functions in the sense of approximating continuous functions in the uniform topology, i.e. how does the Stone-Weierstrass theorem generalize into infinite dimension where measure and compactness are not available?

Let X be a separable real Banach space having a k-times continuously Fréchet differentiable (i.e. Ck-smooth) norm where k∈{1, …, ∞}. We show that any equivalent norm on X can be approximated uniformly on bounded sets by Ck-smooth norms.

We find an optimal upper bound on the values of the weak*-dentability index Dz(X) in terms of the Szlenk index Sz(X) of a Banach space X with separable dual. Namely, if Sz(X) =omega(alpha), for some alpha < omega(1), and p epsilon (1,infinity), then Sz(X) <= Dz(X) <= Sz(L-p(X)) <= {omega(alpha+1) if alpha is a finite ordinal, omega(alpha) if alpha is an infinite ordinal.

Our work is based on a number of tools that are of independent interest. We prove, for every pair of Banach spaces X, Y, that any continuous mapping T : B-X -> Y, which is uniformly differentiable of order up to k in the interior of B-X, can be extended, preserving its best smoothness, into a bidual mapping (T) over tilde : B-X** -> Y**. Another main tool is a result of Zippin's type. We show that weakly Cauchy sequences in X = C(K) can be uniformly well approximated by weakly Cauchy sequences from a certain C[0, alpha], alpha is a countable ordinal, subspace of X**. (C) 2012 Elsevier Inc. All rights reserved.

Banach spaces provide a framework for many branches of mathematics. This book introduces the reader to linear functional analysis and to related parts of infinite-dimensional Banach space theory.

A classical result of Birch claims that for given k, n integers, n-odd there exists some N = N(k, n) such that for an arbitrary n-homogeneous polynomial P on IRN there exists a linear subspace Y hooked right arrow IRN of dimension at least k, where the restriction of P is identically zero (we say that Y is a null space for P). Given n > 1 odd, and arbitrary real separable Banach space X (or more generally a space with w*-separable dual X*), we construct an n-homogeneous polynomial P with the property that for every point 0 not equal x is an element of X there exists some k is an element of IN such that every null space containing x ha's dimension at most k. In particular, P has no infinite dimensional null space. For a given n odd and a cardinal tau, we obtain a cardinal N = N(T, n) = exp(n+1) tau such that every n-homogeneous polynomial on a real Banach space X of density N has a null space of density tau.

It is known that, given a Banach space (X, parallel to center dot parallel to), the modulus of convexity associated to this space delta X is a non-negative function, nondecreasing, bounded above by the modulus of convexity of any Hilbert space and satisfies the equation delta x(epsilon)/epsilon(2) <= 4L delta x(mu)/mu(2) for every 0 < epsilon <= mu <= 2, where L > 0 is a constant. We show that, given a function f satisfying these properties then, there exists a Banach space in such a way its modulus of convexity is equivalent to f, in Figiel's sense. Moreover this Banach space can be taken to be two-dimensional.

Let X be a Banach space, K be a scattered compact and T : BC (K) -> X be a Frechet smooth operator whose derivative is uniformly continuous. We introduce the smooth biconjugate T** : B-C(K)** -> X** and prove that if T is noncompact, then the derivative of T** at some point is a noncompact linear operator. Using this we conclude, among other things, that either <(T(B-c0))over bar> is compact or that l(1) is a complemented subspace of X*. We also give some relevant examples of smooth functions and operators, in particular, a C-1,(u)-smooth noncompact operator from Boo which does not fix any (affine) basic sequence.

We show that the James tree space JT can be renormed to be Lipschitz separated. This negatively answers the question of J. Borwein, J. Giles and J. Vanderwerff as to whether every Lipschitz separated Banach space is an Asplund space.

Let Y be a Banach space, 1 < p < infinity. We give a simple criterion for embedding l(p) subset of Y, namely it suffices that the positive core l(p)(+) subset of Y. This result is applied to the study of highly smooth operators from lp into Y (p is not an even integer). The main result is that every such operator has a harmonic behaviour unless l(p/k) subset of Y for some K is an element of N.

We show that every Frechet differentiable real function on c(0) with locally uniformly continuous derivative has locally compact derivative. Among the corollaries we obtain that there exists no surjective C-2 smooth operator from c(0) onto an infinite dimensional space with nontrivial type.

We show that if the derivative of a convex function on c(0) is locally uniformly continuous, then every point x is an element of c(0) has a neighbourhood O such that f'(O) is relatively compact in l(1).

A new smooth variational principle for spaces admitting Frechet differentiable bump functions is proved. Further it is shown that each proper lower semicontinuous bounded below function can be supported by a smooth function with locally Holder derivative if and only if the space is superreflexive. Some geometrical refinements of the Borwein-Preiss smooth variational principle using Deville's techniques are obtained. (C) 1996 Academic Press, Inc.

A characterisation is given of separable Banach spaces with symmetric Schauder bases which admit equivalent symmetric norms that are Gateaux differentiable or uniformly rotund in every direction. Some applications to questions on distortion of norms on l(infinity) are discussed.