Related papers: Smooth and polyhedral norms via fundamental biorth…

We show that norms on certain Banach spaces $X$ can be approximated uniformly, and with arbitrary precision, on bounded subsets of $X$ by $C^{\infty}$ smooth norms and polyhedral norms. In particular, we show that this holds for any…

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…

In the first part of our note we prove that every Weakly Lindel\"of 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…

Let us consider a Banach space $X$ with the property that every real-valued Lipschitz function $f$ can be uniformly approximated by a Lipschitz, $C^1$-smooth function $g$ with $\Lip(g)\le C \Lip(f)$ (with $C$ depending only on the space…

Let $X$ be a WCG Banach space admitting a $C^k$-Fr\' echet smooth norm. Then $X$ admits an equivalent norm which is simultaneously $C^1$-Fr\' echet smooth, LUR, and a uniform limit of $C^k$-Fr\' echet smooth norms. If $X=C([0,\alpha])$,…

For $1\leq p<\infty$, we prove that the dense subspace $\mathcal{Y}_p$ of $\ell_p(\Gamma)$ comprising all elements $y$ such that $y \in \ell_q(\Gamma)$ for some $q \in (0,p)$ admits a $C^{\infty}$-smooth norm which locally depends on…

We prove two theorems about differentiable functions on the Banach space C(K), where K is compact. (i) If C(K) admits a non-trivial function of class C^m and of bounded support, then all continuous real-valued functions on C(K) may be…

Let $X$ be a Banach space with a separable dual $X^{*}$. Let $Y\subset X$ be a closed subspace, and $f:Y\to\mathbb{R}$ a $C^{1}$-smooth function. Then we show there is a $C^{1}$ extension of $f$ to $X$.

Our note is a complement to recent articles \cite{JS1} (2011) and \cite{JS2} (2013) by M. Jim\'enez-Sevilla and L. S\'anchez-Gonz\'alez which generalise (the basic statement of) the classical Whitney extension theorem for $C^1$-smooth real…

We focus on the presence of almost automorphy in strongly monotone skew-product semiflows on Banach spaces. Under the $C^1$-smoothness assumption, it is shown that any linearly stable minimal set must be almost automorphic. This extends the…

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…

We construct infinitely differentiable norms and partitions of unity for a class of Banach spaces which includes all spaces $\C(K)$ with $K$ a countable compact space, and all spaces $\C_0[0,\Omega )$ with $\Omega $ an ordinal.

Let X be a separable real Banach space having a k-times continuously Fr\'{e}chet 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…

It is shown that a Banach space with locally uniformly convex dual admits an equivalent norm which is itself locally uniformly convex. It follows that on any such space all continuous real-valued functions may be uniformly approximated by…

Our paper is a complement to a recent article by D. Azagra and C. Mudarra (2021). We show how older results on semiconvex functions with modulus $\omega$ easily imply extension theorems for $C^{1,\omega}$-smooth functions on super-reflexive…

We present an example in ZFC of a locally compact, scattered Hausdorff non-Gruenhage space $D$ having a $\G_delta$-diagonal. This answers a question posed by Orihuela, Troyanski and the author in a study of strictly convex norms on Banach…

We study the set $\operatorname{SNA}(M,Y)$ of those Lipschitz maps from a (complete pointed) metric space $M$ to a Banach space $Y$ which (strongly) attain their Lipschitz norm (i.e.\ the supremum defining the Lipschitz norm is a maximum).…

We prove a commutative Gelfand--Naimark type theorem, by showing that the set $C_s(X)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space $X$ (provided with the supremum…

The goal of this note is to prove that every real-valued Lipschitz function on a Banach space can be pointwise approximated on a given $\sigma$-compact set by smooth cylindrical functions whose asymptotic Lipschitz constants are controlled.…

We study $C$-rich spaces, lush spaces, and $C$-extremely regular spaces concerning with the Mazur-Ulam property. We show that a uniform algebra and the real part of a uniform algebra with the supremum norm are $C$-rich spaces, hence lush…