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Petr Hájek

Publications and source records attributed to Petr Hájek.

29 records · Page 2Linked to original sources
arXiv2020

On densely isomorphic normed spaces

In the first part of our note we prove that every Weakly Lindelöf 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. Our main result (Theorem B) concerns the existence of biorthogonal systems in normed spaces. In particular, we prove under the Continuum Hypothesis (CH) that there exists a dense linear subspace of $\ell_2(ω_1)$ (or more generally every WLD space of density $ω_1$) which contains no uncountable biorthogonal system. This result lies between two fundamental results concerning biorthogonal systems, namely the construction of Kunen (under CH) of a non-separable Banach space which contains no uncountable biorthogonal system, and the construction of Todorucević (under Martin Maximum) of an uncountable biorthogonal system in every non-separable Banach space.

math.FA↗
arXiv2020

Almost square dual Banach spaces

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 $\ell_\infty.$ 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.

math.FA↗
arXiv2020

Smooth norms in dense subspaces of Banach spaces

In the first part of our paper, we show that $\ell_\infty$ has a dense linear subspace which admits an equivalent real analytic norm. As a corollary, every separable Banach space, as well as $\ell_1(\mathfrak{c})$, also has a dense linear subspace which admits an analytic renorming. By contrast, no dense subspace of $c_0(ω_1)$ admits an analytic norm. In the second part, we prove (solving in particular an open problem of Guirao, Montesinos, and Zizler) that every Banach space with a long unconditional Schauder basis contains a dense subspace that admits a $C^{\infty}$-smooth norm. Finally, we prove that there is a proper dense subspace of $\ell_{\infty}^{c}(ω_1)$ that admits no Gâteaux smooth norm. (Here, $\ell_{\infty}^{c} (ω_1)$ denotes the Banach space of real-valued, bounded, and countably supported functions on $ω_1$.)

math.FA↗
arXiv2017

CzeV - The Czech Variable Star Catalogue

We present the first release of the Czech Variable star catalogue that currently contains 1228 stars whose variability was discovered by 60 Czech observers. The catalogue contains confirmed variable stars of various types, but also candidates. We give precise coordinates, cross identification with other catalogues, information about constellation, variability type, brightness, light elements, name of the discoverer and year of discovery. In eighty-eight percent of stars the variability type is estimated, for more than 60 % of the stars the light ephemerides are given.

astro-ph.SR↗
arXiv2017

Strongly extreme points and approximation properties

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. In contrast to the above results we also construct a non-symmetric norm on $c_0$ for which all points on the unit sphere are strongly extreme, but none of these points are denting.

math.FA↗
arXiv2017

Some remarks on the structure of Lipschitz-free spaces

We give several structural results concerning the Lipschitz-free spaces $\mathcal F(M)$, where $M$ is a metric space. We show that $\mathcal F(M)$ contains a complemented copy of $\ell_1(Γ)$, where $Γ=\text{dens}(M)$. If $\mathcal N$ is the net in a finite dimensional Banach space $X$, we show that $\mathcal F(\mathcal N)$ is isomorphic to its square. If $X$ contains a complemented copy of $\ell_p, c_0$ then $\mathcal F(\mathcal N)$ is isomorphic to its $\ell_1$-sum. Finally, we prove that for all $X\cong C(K)$ spaces $\mathcal F(\mathcal N)$ are mutually isomorphic spaces with a Schauder basis.

math.FA↗
arXiv2015

On uniformly differentiable mappings from $\ell_\infty(Γ)$

In 1970 Haskell Rosenthal proved that if $X$ is a Banach space, $Γ$ is an infinite index set, and $T:\ell_\infty(Γ)\to X$ is a bounded linear operator such that $\inf_{γ\inΓ}\|T(e_γ)\|>0$ then $T$ acts as an isomorphism on $\ell_\infty(Γ')$, for some $Γ'\subsetΓ$ 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 ${F}:B_{\ell_\infty(Γ)}\to X$ is uniformly differentiable and such that $\inf_{γ\inΓ}\|{F}(e_γ){-F(0)}\|>0$ then there exists $x\in B_{\ell_\infty(Γ)}$ such that $d{F}(x)[\cdot]$ is a bounded linear operator which acts as an isomorphism on $\ell_\infty(Γ')$, for some $Γ'\subsetΓ$ of the same cardinality as $Γ$.

math.FA↗
arXiv2015

Approximation and Schur properties for Lipschitz free spaces over compact metric spaces

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.

math.FA↗
arXiv2012

Smooth approximations of norms in separable Banach spaces

Let X be a separable real Banach space having a k-times continuously Fréchet differentiable (i.e. C^k-smooth) norm where k=1,...,\infty. We show that any equivalent norm on X can be approximated uniformly on bounded sets by C^k-smooth norms.

math.FA↗
arXiv2009

On Peano's theorem in Banach spaces

We show that if $X$ is an infinite-dimensional separable Banach space (or more generally a Banach space with an infinite-dimensional separable quotient) then there is a continuous mapping $f\colon X\to X$ such that the autonomous differential equation $x'=f(x)$ has no solution at any point.

math.CA↗