论文标题
不对称的自由空间和规范不对称
Asymmetric free spaces and canonical asymmetrizations
论文作者
论文摘要
类似于戈德罗伊 - 卡尔顿（Godefroy-Kalton）的公制空间的结构允许以规范的方式嵌入每个准米度空间$（x，d）$中，以一种不对称的规范空间$ \ mathcal {f} _a（x，x，d）$（其quasi-metric-metric-metric-metric免费空间，也称为免费的空间）。准 - 金属自由空间满足通用性能（半lipschitz函数的线性化）。 The (conic) dual of $\mathcal{F}_a(X,d)$ coincides with the nonlinear asymmetric dual of $(X,d)$, that is, the space $\mathrm{SLip}_0(X,d)$ of semi-Lipschitz functions on $(X,d)$, vanishing at a base point.特别是，对于公制空间$（x，d）$，上述构造可产生通常的自由空间。另一方面，每个度量空间$（x，d）$自然而然地继承了一个规范的不对称化，来自其自由空间$ \ Mathcal {f}（x）$。这引起了准米特空间$（x，d _+）$和不对称的免费空间$ \ MATHCAL {f} _a（x，d _+）$。后者的对称性与原始自由空间$ \ Mathcal {f}（x）$是同构。用明确的例子说明了这项工作的结果。
A construction analogous to that of Godefroy-Kalton for metric spaces allows to embed isometrically, in a canonical way, every quasi-metric space $(X,d)$ to an asymmetric normed space $\mathcal{F}_a(X,d)$ (its quasi-metric free space, also called asymmetric free space or semi-Lipschitz free space). The quasi-metric free space satisfies a universal property (linearization of semi-Lipschitz functions). The (conic) dual of $\mathcal{F}_a(X,d)$ coincides with the nonlinear asymmetric dual of $(X,d)$, that is, the space $\mathrm{SLip}_0(X,d)$ of semi-Lipschitz functions on $(X,d)$, vanishing at a base point. In particular, for the case of a metric space $(X,D)$, the above construction yields its usual free space. On the other hand, every metric space $(X,D)$ inherits naturally a canonical asymmetrization coming from its free space $\mathcal{F}(X)$. This gives rise to a quasi-metric space $(X,D_+)$ and an asymmetric free space $\mathcal{F}_a(X,D_+)$. The symmetrization of the latter is isomorphic to the original free space $\mathcal{F}(X)$. The results of this work are illustrated with explicit examples.
