A. Assadi, Hadi Haghshenas, H. Hosseini Guive
In this paper, we prove the Smulian s theorem on Frechet differentiability of norm,and present some of its geometric results concerning the Gateaux and Frechet differentiability of norm and properties of the allied space and its dual such as reflexivity and strict convexity.
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Strict Frechet and generalized differentiability of the metric projection operator onto balls in Hilbert spaces
本文证明了Hilbert空间中到闭球的度量投影算子的严格Frechet可微性并求出导数表达式，强化了已有方向可微性结果，并据此研究了该算子的广义可微性。
Strict Frechet Differentiability of the Metric Projection Operator in Hilbert Spaces
本文证明希尔伯特空间闭球和欧几里得空间正锥上度量投影算子的严格弗雷歇可微性，给出导数精确表达式，从而加强了已有方向可微性结果。
Some Results on the Strict Fréchet Differentiability of the Metric Projection Operator in Hilbert Spaces
本文给出了Hilbert空间中到闭球及欧几里得空间中到二阶锥的度量投影算子严格Fréchet可微性的简洁证明与推广。
Let $X_1, \dots, X_n$ be Banach spaces and $f$ a real function on $X=X_1 \times\dots \times X_n$. Let $A_f$ be the set of all points $x \in X$ at which $f$ is partially Fr\' echet differentiable but is not Fr\' echet differentiable. Our results imply that if $X_1, \dots, X_{n-1}$ are Asplund spaces and $f$ is continuous (resp. Lipschitz) on $X$, then $A_f$ is a first category set (resp. a $σ$-upper porous set). We also prove that if $X$, $Y$ are separable Banach spaces and $f:X \to Y$ is a Lipschitz mapping, then the set of all points $x \in X$ at which $f$ is G\^ ateaux differentiable, is Fr\…
Caracterisation of Frechet differentiability norm for dual complex Banach spaces
本文刻画了复Banach空间对偶空间的Frechet可微性，并证明若X的对偶空间Gateaux可微，则Lp(X)的对偶空间对所有p同样具备该性质。
Frechet differentiability of the metric projection operator in Banach spaces
本文研究了一致凸且一致光滑的 Banach 空间中度量投影算子的 Frechet 可微性，并针对闭球、闭凸圆柱和正锥等闭凸子集，求得了其 Frechet 导数与 Gateaux 方向导数的精确表达式。
Frechet differentiability and quasi-polyhedrality in spaces of operators
本文研究了在算子空间中非常光滑点的弗雷歇可微性，揭示了高阶偶数次对偶空间范数光滑性与算子空间光滑性的关系。
We consider convex series of molecules in Lipschitz-free spaces, i.e. elements of the form $μ=\sum_n λ_n \frac{δ_{x_n}-δ_{y_n}}{d(x_n,y_n)}$ such that $\|μ\|=\sum_n |λ_n |$. We characterise these elements in terms of geometric conditions on the points $x_n$, $y_n$ of the underlying metric space, and determine when they are points of Gâteaux differentiability of the norm. In particular, we show that Gâteaux and Fréchet differentiability are equivalent for finitely supported elements of Lipschitz-free spaces over uniformly discrete and bounded metric spaces, and that their tensor products with G…
One of the long standing questions in the theory of Schatten-von Neumann ideals of compact operators is whether their norms have the same differentiability properties as the norms of their commutative counterparts. We answer this question in the affirmative. A key technical observation underlying our proof is a discovery of connection between this question and recent affirmative resolution of L.S. Koplienko's conjecture concerning existence of higher order spectral shift functions.
We establish the following results on higher order $\mathcal{S}^p$-differentiability, $1<p<\infty$, of the operator function arising from a continuous scalar function $f$ and self-adjoint operators defined on a fixed separable Hilbert space: (i) $f$ is $n$ times continuously Fréchet $\mathcal{S}^p$-differentiable at every bounded self-adjoint operator if and only if $f\in C^n(\mathbb{R})$; (ii) if $f',\ldots,f^{(n-1)}\in C_b(\mathbb{R})$ and $f^{(n)}\in C_0(\mathbb{R})$, then $f$ is $n$ times continuously Fréchet $\mathcal{S}^p$-differentiable at every self-adjoint operator; (iii) if $f',\ldot…
We study the relationships between Gateaux, weak Hadamard and Frechet differentiability and their bornologies for Lipschitz and for convex functions. In particular, Frechet and weak Hadamard differentiabily coincide for all Lipschitz functions if and only if the space is reflexive (an earlier paper of the first two authors shows that these two notions of differentiability coincide for continuous convex functions if and only if the space does not contain a copy of $\ell_1$). We also examine when Gateaux and weak Hadamard differentiability coincide for continuous convex functions. For instance, …
the main goal of this paper is to prove that any Banach space X, that every dual ball in X** is weak* -separable, or every weak* -closed convex subset in X** is weak* -separable, or every norm-closed convex set in X* is constructible, admits an equivalent Frechet differentiable norm.
Differentiability of the operator norm on $\ell_p$ spaces
研究\(\ell_p\)空间上有界线性算子范数的可微性，给出强次可微性刻画、特定集合稠密性证明及弗雷歇可微性刻画，证明了该空间上算子范数的弗雷歇可微性与加托可微性一致，推广了希尔伯特空间相关定理。
We prove an implicit function theorem for Keller C^k_c-maps from arbitrary real or complex topological vector spaces to Frechet spaces, imposing only a certain metric estimate on the partial differentials. As a tool, we show the C^k-dependence of fixed points on parameters for suitable families of contractions of a Frechet space. The investigations were stimulated by a recent metric approach to differentiability in Frechet spaces by Olaf Mueller. Our results also subsume generalizations of Mueller's Inverse Function Theorem for mappings between Frechet spaces. As an application, we prove exist…
The first Fundamental Theorem of Calculus for functions defined on Wasserstein space
本文在Wasserstein空间上建立了微积分基本定理的类比，证明了足够规则的函数积分可微且导数与被积函数一致。
Let $K$ be a Hausdorff space and $C_b(K)$ be the Banach algebra of all complex bounded continuous functions on $K$. We study the Gâteaux and Fréchet differentiability of subspaces of $C_b(K)$. Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace $H$ of $C_b(K)$ is a dense $G_δ$ subset of $H$, if $K$ is compact. This gives a generalized Bishop's theorem, which says that the closure of the set of strong peak point for $H$ is the smallest closed norming subset of $H$. The classical Bishop's theorem was proved for a separating subalgebra $H$ a…
For the Frechet space E=C^{\infty}(S^1) and for a smooth ϕ: R to R, we prove that the associated map E to E given by x mapstoϕ\circ x satisfies the continuous BΓ--differentiability condition in Yamamuro's inverse function theorem only if ϕis affine. Via more complicated examples, we also generally discuss the importance of testing the applicability of proposed inverse and implicit function theorems by this kind of simple maps.
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