Quotients of finite-dimensional quasi-normed spaces

Statistical quasi Cauchyness in two normed spaces

A function $f$ defined on a subset $E$ of a two normed space $X$ is statistically ward continuous if it preserves statistically quasi-Cauchy sequences of points in $E$ where a sequence $(x_n)$ is statistically quasi-Cauchy if $(Δx_{n})$ is a statistically null sequence. A subset $E$ of $X$ is statistically ward compact if any sequence of points in $E$ has a statistically quasi-Cauchy subsequence. In this paper, new kinds of continuities are investigated in two normed spaces. It turns out that uniform limit of statistically ward continuous functions is again statistically ward continuous.

拟范数群中双边逼近的若干精确常数

Some exact constants for bilateral approximations in quasi-normed groups

本文在拟范数阿贝尔群中利用带精确常数的双边 Bernstein-Jackson 不等式建立最佳逼近的逆定理与正定理，并通过递减重排和 E-泛函给出误差估计及数值与谱逼近实例。

Factorization theorems for quasi-normed spaces

We extend Pisier's abstract version of Grothendieck's theorem to general non-locally convex quasi-Banach spaces. We also prove a related result on factoring operators through a Banach space and apply our results to the study of vector-valued inequalities for Sidon sets. We also develop the local theory of (non-locally convex) spaces with duals of weak cotype 2.

On sequences in 2-normed spaces

A function $f$ defined on a 2-normed space $ (X,||.,.||)$ is ward continuous if it preserves quasi-Cauchy sequences where a sequence $(x_n)$ of points in $X$ is called quasi-Cauchy if $lim_{n\rightarrow\infty}||Δx_{n},z||=0$ for every $z\in X$. Some other kinds of continuties are also introduced via quasi-Cauchy sequences in 2-normed spaces. It turns out that uniform limit of ward continuous functions is again ward continuous.

Quasi-modular spaces with applications to quasi-normed Calderón-Lozanovski\uı spaces

In this paper we introduce the notion of a quasi-modular and we prove that the respective Minkowski functional of the unit quasi-modular ball becomes a quasi-norm. In this way, we refer to and complete the well-known theory related to the notions of a modular and a convex modular that lead to the F-norm and to the norm, respectively. We use the obtained results to consider basic properties of quasi-normed Calderón-Lozanovski\uı spaces $E_φ$, where the lower Matuszewska-Orlicz index $α_φ$ plays the key role. We also give a number of theorems concerning different copies of $l^{\infty}$ in the sp…

Saturating Constructions for Normed Spaces

We prove several results of the following type: given finite dimensional normed space V there exists another space X with log (dim X) = O(log (dim V)) and such that every subspace (or quotient) of X, whose dimension is not "too small," contains a further subspace isometric to V. This sheds new light on the structure of such large subspaces or quotients (resp., large sections or projections of convex bodies) and allows to solve several problems stated in the 1980s by V. Milman. The proofs are probabilistic and depend on careful analysis of images of convex sets under Gaussian linear maps.

A study involving the completion of quasi 2-normed space

The fundamental aim of this paper is to introduce and investigate a new property of quasi 2-normed space based on a question given by C. Park (2006) [2] for the completion quasi 2-normed space. Finally, we also find an answer for a question Park's.

Tensor products of normed and Banach quasi *-algebras

Quasi *-algebras form an essential class of partial *-algebras, which are algebras of unbounded operators. In this work, we aim to construct tensor products of normed, respectively Banach quasi *-algebras, and study their capacity to preserve some important properties of their tensor factors, like for instance, *-semisimplicity and full representability.

$M$-ideals and split faces of the quasi state space of a non-unital ordered Banach space

We characterize $M$-ideals in order smooth $\infty$-normed spaces by extending the notion of split faces of the state space to those of the quasi-state space. We also characterize approximate order unit spaces as those order smooth $\infty$-normed spaces $V$ that are $M$-ideals in $\tilde{V}.$ Here $\tilde{V}$ is the order unit space obtained by adjoining an order unit to $V.$ To prove these results, we develop an order theoretic version of the "Alfsen-Efffros' cone decomposition theorem" for order smooth $1$-normed spaces. (As a quick application of this result, we sharpen a result on the ext…

Hahn-Banach theorems for $κ$-normed spaces

For a new class of topological vector spaces, namely $κ$-normed spaces, and associated quasisemilinear topological preordered space is defined and investigated. This structure arise naturally from the consideration of a $κ$-norm, that is a distance function between a point and a $G_δ$-subset. For it, analogs of the Hahn-Banach theorem are proved.

非对称赋范空间上线性算子的Hahn-Banach型延拓结果

Hahn-Banach type extension results for linear operators on asymmetric normed spaces

本文针对非对称赋范空间上线性算子的延拓问题，将Nachbin的经典思路适配到非对称情形，给出了混合二元交性质与非对称赋范空间单射性等价这一已有结论的直接证明，丰富了非对称赋范空间下的Hahn-Banach型理论。

Local geometric properties in quasi-normed Orlicz spaces

Several local geometric properties of Orlicz space $L_ϕ$ are presented for an increasing Orlicz function $ϕ$ which is not necessarily convex, and thus $L_ϕ$ does not need to be a Banach space. In addition to monotonicity of $ϕ$ it is supposed that $ϕ(u^{1/p})$ is convex for some $p>0$ which is equivalent to that its lower Matuszewska-Orlicz index $α_ϕ>0$. Such spaces are locally bounded and are equipped with natural quasi-norms. Therefore many local geometric properties typical for Banach spaces can also be studied in those spaces. The techniques however have to be different, since duality the…

Saturating Constructions for Normed Spaces II

We prove several results of the following type: given finite dimensional normed space V possessing certain geometric property there exists another space X having the same property and such that (1) log (dim X) = O(log (dim V)) and (2) every subspace of X, whose dimension is not "too small," contains a further well-complemented subspace nearly isometric to V. This sheds new light on the structure of large subspaces or quotients of normed spaces (resp., large sections or linear images of convex bodies) and provides definitive solutions to several problems stated in the 1980s by V. Milman. The pr…

The non-compact normed space of norms on a finite-dimensional Banach space

We discuss a new pseudometric on the space of all norms on a finite-dimensional vector space (or free module) $\mathbb{F}^k$, with $\mathbb{F}$ the real, complex, or quaternion numbers. This metric arises from the Lipschitz-equivalence of all norms on $\mathbb{F}^k$, and seems to be unexplored in the literature. We initiate the study of the associated quotient metric space, and show that it is complete, connected, and non-compact. In particular, the new topology is strictly coarser than that of the Banach-Mazur compactum. For example, for each $k \geqslant 2$ the metric subspace $\{ \| \cdot \…

The normed and Banach envelopes of Weak L^1

The space Weak L^1 consists of all measurable functions on [0,1] such that q(f) = sup_{c>0} c λ{t : |f(t)| > c} is finite, where λdenotes Lebesgue measure. Let ρbe the gauge functional of the unit ball {f : q(f) \leq 1} of the quasi- norm q, and let N be the null space of ρ. The normed envelope of Weak L^1, which we denote by W, is the space (Weak L^1/N, ρ). The Banach envelope of Weak L^1, \overline{W}, is the completion of W. We show that \overline{W} is isometrically lattice isomorphic to a sublattice of W. It is also shown that all rearrangement invariant Banach function spaces are isometr…

关于Daugavet空间的光滑和非贫困子空间

About smooth and non-poor subspaces of Daugavet spaces

研究非完备范数空间中具有Daugavet性质的光滑范数，以及其对偶范数的非光滑性，并探讨Quasilacunary Müntz空间作为C[0,1]的子空间的性质。

A variation of continuity in $n$-normed spaces

The s-th forward difference sequence that tends to zero, inspired by the consecutive terms of a sequence approaching zero, is examined in this study. Functions that take sequences satisfying this condition to sequences satisfying the same condition are called s-ward continuous. Inclusion theorems that are related to this kind of uniform continuity and continuity are also considered. Additionally, the concept of $s$-ward compactness of a subset of $X$ via $s$-quasi-Cauchy sequences are investigated. One finds out that the uniform limit of any sequence of $s$-ward continuous function is $s$-ward…

有界多线性算子的$L^p_μ$赋范空间

$L^p_μ$-Normed Spaces of Bounded Multilinear Operator

该研究探讨配备$L^p_μ$范数的有界多线性算子空间，明确其诱导拓扑的一致性，并确定不同$p$值下空间完备化的对应算子空间，丰富了赋范空间理论的相关内容。

非对称赋范实向量空间的Isbell-凸包上的提升Takahashi凸性

Lifted Takahashi Convexity on the Isbell-Convex Hull of an Asymmetrically Normed Real Vector Space

本文在非对称赋范实向量空间的Isbell-凸包上定义了一种提升凸性结构，并证明了其构成一个凸T0-拟度量空间，同时与规范嵌入兼容且与最小函数对的W-凸性相关。

随机范数模的Bochner型积分理论

A Bochner-type integration theory for random normed modules

本文发展了随机范数模的测度与积分理论，研究了值域为L^0(m)的测度，并建立了针对完整随机范数模的Bochner型积分理论，证明了Radon-Nikodým定理和Riesz-Markov-Kakutani表示定理的版本，并探讨了其应用。

与定义在赋范空间乘积的开集上、取值于拟完备局部凸线性空间的拉格朗日量相关联的普适算子诺特流

Universal operatorial Noether current associated with a quasi-complete locally convex linear space valued Lagrangian defined on an open set of a product of normed spaces

本文为局部凸向量值拉格朗日量建立变分框架，构造普适算子诺特流，证明其壳上守恒及无穷阶时的散度为零。

Variants of the Maurey-Rosenthal theorem for quasi K"othe function spaces

The Maurey-Rosenthal theorem states that each bounded and linear operator T from a quasi normed space E into some L_p(ν) which satisfies a certain vector-valued inequality even allows a weighted norm inequality. Continuing the work of Garcia Cuerva and Rubio de Francia we give several scalar and vector-valued variants of this fundamental result within the framework of quasi K"othe function spaces over measure spaces.

Isometric representation of Lipschitz-free spaces over convex domains in finite-dimensional spaces

Let $E$ be a finite-dimensional normed space and $Ω$ a nonempty convex open set in $E$. We show that the Lipschitz-free space of $Ω$ is canonically isometric to the quotient of $L^1(Ω,E)$ by the subspace consisting of vector fields with zero divergence in the sense of distributions on $E$.

On Weak Supercyclicity II

This paper considers weak supercyclicity for bounded linear operators on a normed space. On the one hand, weak supercyclicity is investigated for classes of Hilbert-space operators: (i) self-adjoint operators are not weakly supercyclic, (ii) diagonalizable operators are not weakly l-sequentially supercyclic, and (iii) weak l-sequential supercyclicity is preserved between a unitary operator and its adjoint. On the other hand, weak supercyclicity is investigated for classes of normed-space operators: (iv) the point spectrum of the normed-space adjoint of a power bounded supercyclic operator is e…

Certain properties of Generalization of $L^p-$Spaces for $0 < p < 1$

This paper introduces the notion of $N^*-$function and gives a generalization of $L^p,$ for $0<p<1$ denoted by $L_Φ$ where $Φ$ is an $N^*-$function. As well as, this paper examines some properties regarding to this generalized spaces and its linear forms, including some analogies and common features to some other well known spaces. As well as, we prove this space is a quasi-normed space but it is not normed space.

Open quotients of trivial vector bundles

Given an arbitrary topological complex vector space $A$, a quotient vector bundle for $A$ is a quotient of a trivial vector bundle $π_2:A\times X\to X$ by a fiberwise linear continuous open surjection. We show that this notion subsumes that of a Banach bundle over a locally compact Hausdorff space $X$. Hyperspaces consisting of linear subspaces of $A$, topologized with natural topologies that include the lower Vietoris topology and the Fell topology, provide classifying spaces for various classes of quotient vector bundles, in a way that generalizes the classification of locally trivial vector…

Quasi-algebra, a special sample of quasilinear spaces

Similar to linear spaces, many examples of quasilinear spaces have a notion of multiplication of the elements. To characterising these examples, in the present paper we generalize the notion of quasilinear spaces and introduce quasi-algebras and normed quasi-algebras with the help of what was done in constructing algebras. After examining some properties of these new spaces, we introduce the concept of quasi-homomorphisms that is an instance of a quasilinear operator. Moreover since the study of spaces with algebraic structures cannot be completed without the study of spectrums, we define the …

On non-linear mappings preserving the semi-inner product

We say that a smooth normed space $X$ has a property (SL), if every mapping $f:X \to X$ preserving the semi-inner product on $X$ is linear. It is well known that every Hilbert space has the property (SL) and the same is true for every finite-dimensional smooth normed space. In this paper, we establish several new results concerning the property (SL). We give a simple example of a smooth and strictly convex Banach space which is isomorphic to the space $\ell_p$, but without the property (SL). Moreover, we provide a characterization of the property (SL) in the class of reflexive smooth Banach sp…

Stability of Euler-Lagrange type cubic functional equations in quasi-Banach spaces

In this paper, we study the generalized Hyers-Ulam stability of Euler-Lagrange type cubic functional equation of the form \begin{align*} 2mf(x + my) + 2f(mx - y) = (m^3 + m)[f(x+ y) + f(x - y)] + 2(m^4 - 1)f(y) \end{align*} for all $x,y \in X$, where $m$ is a fixed scalar such that $m \neq 0,1$, and $f$ is a map from a quasi-normed space $X$ to a quasi-Banach space $Y$ over the same field with $X$ by applying the alternative fixed point theorem.

On the minimal space problem and a new result on existence of basic sequences in quasi-Banach spaces

We prove that if $X$ is a quasi-normed space which possesses an infinite countable dimensional subspace with a separating dual, then it admits a strictly weaker Hausdorff vector topology. Such a topology is constructed explicitly. As an immediate consequence, we obtain an improvement of a well-known result of Kalton-Shapiro and Drewnowski by showing that a quasi-Banach space contains a basic sequence if and only if it contains an infinite countable dimensional subspace whose dual is separating. We also use this result to highlight a new feature of the minimal quasi-Banach space constructed by …

乘法算子的范数：回答Fialkow-Loebl问题

Norms of multiplication operators: answering Fialkow--Loebl question

该研究针对因子上的乘法算子，证明其值域包含于对称赋范算子空间的充要条件，给出其范数公式并回答了Fialkow-Loebl问题，还分析了弱$L_p$空间自然拟范数的非单调性。

Compact operators on spaces with asymmetric norm

The aim of the present paper is to define compact operators on asymmetric normed spaces and to study some of their properties. The dual of a bounded linear operator is defined and a Schauder type theorem is proved within this framework. The paper contains also a short discussion on various completeness notions for quasi-metric and for quasi-uniform spaces.

非可逆Finsler流形间等距映射的泛函刻画

A functional characterization of isometries between non-reversible Finsler manifolds

本文通过推广Myers-Nakai定理，利用光滑半Lipschitz函数锥及新的代数/拟度量结构，完整刻画了非可逆Finsler流形间的等距映射。

无限余维子空间的插值

Interpolation of subspaces of infinite codimension

本文研究无限余维闭子空间替换后插值空间的比较问题，提出商 J 范数作为处理多种相关情形的有效工具。

度量空间并集与商集上的 Lipschitz 函数

Lipschitz Functions on Unions and Quotients of Metric Spaces

本文基于 Pelczyński 分解方法和几何方法，证明了有限个具有有限 Nagata 维数且 Lipschitz 自由空间同构于 $L^1$ 的度量空间并集（包括拟共形树及其度量商集）的 Lipschitz 自由空间同构于 $L^1$，并确定其 Lipschitz 维数为 1。

赋范空间有限子集的Lipschitz收缩研究

On Lipschitz Retraction of Finite Subsets of Normed Spaces

该研究针对赋范空间有限子集空间的Lipschitz收缩问题，证明了多类收缩映射的存在性与正则性，还得到测地度量空间有限子集空间的2-拟凸性，推进了巴拿赫空间相关收缩问题的研究。

准均匀和准模空间中的局部反称连通性

Local Antisymmetric Connectedness in Quasi-Uniform and Quasi-Modular Spaces

本文研究了准均匀和准模伪度量空间中的反称连通性，建立了相关特征化和稳定性结果，并探讨了其与Smyth完备性和Yoneda型完备性的联系。

Similarity of Quotient Hilbert modules in the Cowen-Douglas Class

In this paper, we consider the similarity and quasi-affinity problems for Hilbert modules in the Cowen-Douglas class associated with the complex geometric objects, the hermitian anti-holomorphic vector bundles and curvatures. Given a "simple" rank one Cowen-Douglas Hilbert module $\mathcal{M}$, we find necessary and sufficient conditions for a class of Cowen-Douglas Hilbert modules satisfying some positivity conditions to be similar to $\mathcal{M} \otimes \mathbb{C}^m$. We also show that under certain uniform bound condition on the anti-holomorphic frame, a Cowen-Douglas Hilbert module is qua…

On extremal sections of subspaces of $L_p$

Let $m,n\in\mathbb{N}$ and $p\in(0,\infty)$. For a finite dimensional quasi-normed space $X=(\mathbb{R}^m, \|\cdot\|_X)$, let $$B_p^n(X) = \Big\{ (x_1,\ldots,x_n)\in\big(\mathbb{R}^{m}\big)^n: \ \sum_{i=1}^n \|x_i\|_X^p \leq 1\Big\}.$$ We show that for every $p\in(0,2)$ and $X$ which admits an isometric embedding into $L_p$, the function $$S^{n-1} \ni θ= (θ_1,\ldots,θ_n) \longmapsto \Big| B_p^n(X) \cap\Big\{(x_1,\ldots,x_n)\in \big(\mathbb{R}^{m}\big)^n: \ \sum_{i=1}^n θ_i x_i=0 \Big\} \Big|$$ is a Schur convex function of $(θ_1^2,\ldots,θ_n^2)$, where $|\cdot|$ denotes Lebesgue measure. In pa…