设$E$是一个无限维可分Hilbert空间。我们证明了对于每个$C^1$函数$f:E\to\mathbb{R}^d$，每个具有$C_f:=\{x\in E:\,Df(x)\; \text{is not surjective}\}\subset U$的开集$U$和每个连续函数$\varepsilon:E\to (0,\infty)$，都存在一个$C^1$映射$φ:E\to\mathbb{R}^d$，使得对于每个$x\in E$，$f=φ$在$U$和$φ$之外的$||f(x)-φ(x)||\leq \varepsilon(x)$没有临界点（$C_φ=\emptyset$）.这一结果可以推广到$E=c_0$或$E=l_p$，$1<p<\infty$的情况。在$E=c_0$的情况下，还可以为每个$x\in E$获得该$||Df(x)-Dφ(x)||\leq\varepsilon(x)$。
英文摘要：
Let $E$ be an infinite-dimensional separable Hilbert space. We show that for every $C^1$ function $f:E\to\mathbb{R}^d$, every open set $U$ with $C_f:=\{x\in E:\,Df(x)\; \text{is not surjective}\}\subset U$ and every continuous function $\varepsilon:E\to (0,\infty)$ there exists a $C^1$ mapping $φ:E\to\mathbb{R}^d$ such that $||f(x)-φ(x)||\leq \varepsilon(x)$ for every $x\in E$, $f=φ$ outside $U$ and $φ$ has no critical points ($C_φ=\emptyset$). This result can be generalized to the case where $E=c_0$ or $E=l_p$, $1<p<\infty$. In the case $E=c_0$ it is also possible to get that $||Df(x)-Dφ(x)||\leq\varepsilon(x)$ for every $x\in E$.
For $1<p\le \infty$, we show the existence of a Banach space which is both injectively and surjectively universal for the class of all separable Banach spaces with an equivalent $p$-asymptotically uniformly smooth norm. We prove that this class is analytic complete in the class of separable Banach spaces. These results extend previous works by Kalton, Werner and Kurka in the case $p=\infty$.
We characterize $k-$smoothness of an element on the unit sphere of a finite-dimensional polyhedral Banach space. Then we study $k-$smoothness of an operator $T \in \mathbb{L}(\ell_{\infty}^n,\mathbb{Y}),$ where $\mathbb{Y}$ is a two-dimensional Banach space with the additional condition that $T$ attains norm at each extreme point of $B_{\ell_{\infty}^{n}}.$ We also characterize $k-$smoothness of an operator defined between $\ell_{\infty}^3$ and $\ell_{1}^3.$
For $1<p\leqslant \infty$, we study the complexity and the existence of universal spaces for two classes of separable Banach spaces, denoted $\textsf{A}_p$ and $\textsf{N}_p$, and related to asymptotic smoothness in Banach spaces. We show that each of these classes is Borel in the class of separable Banach spaces. Then we build small families of Banach spaces that are both injectively and surjectively universal for these classes. Finally, we prove the optimality of this universality result, by proving in particular that none of these classes admits a universal space.
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.
Approximate Carath{é}odory's theorem in uniformly smooth Banach spaces
本文研究了Carath{é}odory定理在巴拿赫空间中的“无量纲”类似。我们在一致光滑巴拿赫空间中证明了这样一个结果及其彩色版本。由此可见，均匀光滑性导致了Maurey的经典引理\cite{pisier1980remarques}的贪婪去随机化，这本身就是Carath{é}odory定理的一个概率证明的“无维”模拟。我们用$1 < p \leq 2$找到了$L_p$中有界集的凸包与$k$ -凸包的偏差的渐近紧上界，并得到了与用$1 < p < \infty.$的$L_p$的Maurey引理中的偏差的渐近紧上界
Real smooth three-dimensional or higher Banach spaces are isomorphic with respect to the nonlinear structure of Birkhoff-James orthogonality if and only if they are isometrically isomorphic. Moreover, using smooth Radon planes and non-smooth direct sums, in arbitrary dimensions, we construct examples of non-isometric pairs of real Banach spaces that admit norm-preserving homogeneous bicontinuous Birkhoff-James orthogonality preservers among them.
Let X be a real uniformly convex and uniformly smooth Banach space and C a nonempty closed and convex subset of X. Let Pc from X to C denote the (standard) metric projection operator. In this paper, we define the Gateaux directional differentiability of Pc. We investigate some properties of the Gateaux directional differentiability of Pc. In particular, if C is a closed ball or a closed and convex cone (including proper closed subspaces), then, we give the exact representations of the directional derivatives of Pc. Finally, we define the concept of p-q uniformly convex and uniformly smooth Ban…
Some remarks on smooth mappings of Hilbert and Banach spaces and their local convexity property
University of Agriculture in Krakow（克拉科夫农业大学）；Institute of Mathematics of NAS of Ukraine（乌克兰国家科学院数学研究所）；Lviv Polytechnic National University（利沃夫国立理工大学）；Comenius University in Bratislava（布拉迪斯拉发康斯坦丁尼查尔大学）
This article develops optimality conditions for a large class of non-smooth variational models. The main results are based on standard tools of functional analysis and calculus of variations. Firstly we address a model with equality constraints and, in a second step, a more general model with equality and inequality constraints, always in a general Banach space context. We highlight the results in general are well known, however, some novelties are introduced related to the proof procedures, which are in general softer than those concerning the present literature.
We develop tools to produce equivalent norms with specific local geometry around infinitely many points in the sphere of a Banach space via an inductive procedure. We combine this process with smoothness results and techniques to solve two open problems posed in the recently published monograph [GMZ22] by A. J. Guirao, V. Montesinos and V. Zizler. Specifically, on the one hand we construct in every separable Banach space admitting a $C^k$-smooth norm an equivalent norm which is $C^k$-smooth but fails to be uniformly Gâteaux in any direction; and on the other hand we produce in $c_0(Γ)$ for any…
We explore Birkhoff-James orthogonality of two elements in a complex Banach space by using the directional approach. Our investigation illustrates the geometric distinctions between a smooth point and a non-smooth point in a complex Banach space. As a concrete outcome of our study, we obtain a new proof of the Bhatia-Šemrl Theorem on orthogonality of linear operators.
A Banach space $X$ has the $Mazur$-$Ulam$ $property$ if any isometry from the unit sphere of $X$ onto the unit sphere of any other Banach space $Y$ extends to a linear isometry of the Banach spaces $X,Y$. A Banach space $X$ is called $smooth$ if the unit ball has a unique supporting functional at each point of the unit sphere. We prove that each non-smooth 2-dimensional Banach space has the Mazur-Ulam property.
This paper presents a survey of maximal inequalities for stochastic convolutions in $2$-smooth Banach spaces and their applications to stochastic evolution equations.
We prove that any isometry between the unit spheres of $C^2$-smooth (more generally, absolutely smooth) smooth Banach spaces extends to a linear isometry of the Banach spaces. This answers the famous Tingley's problem in the class of absolutely smooth $2$-dimensional Banach spaces.
