论文标题
矢量空间上的自由向量晶格作为功能晶格
Free vector lattices over vector spaces as function lattices
论文作者
论文摘要
我们表明，在实际矢量空间$ v $上的自由矢量晶格可以在其双重空间的任何线性子空间上以实现$ v $的点的任何线性子空间上的实现正同质函数来实现。这是为了使已知的事实提供直觉，即可以将Banach Space $ e $的免费Banach晶格实现为在$ e^\ ast $上具有正同质功能的Banach晶格。它还用于改善众所周知的结果，即可以将非空置集的自由向量晶格实现为实值函数的向量晶格。对于无限集，可以选择这种实现的基础空间比通常的空间要小。
We show that a free vector lattice over a real vector space $V$ can be realised canonically as a vector lattice of real-valued positively homogeneous functions on any linear subspace of its dual space that separates the points of $V$. This is used to give intuition for the known fact that the free Banach lattice over a Banach space $E$ can be realised as a Banach lattice of positively homogeneous functions on $E^\ast$. It is also applied to improve the well-known result that free vector lattices over non-empty sets can be realised as vector lattices of real-valued functions. For infinite sets, the underlying spaces for such realisations can be chosen to be smaller than the usual ones.
