Multi-norms and multi-Banach algebras

线性空间是普通n维空间到无限维空间的推广。在这样的空间中，我们可以用范数来测量向量的长度，从而推广了普通n维空间中向量长度的通常概念。这就引出了“赋范线性空间”的公理化定义；如果我们的空间有一个额外的“完备性”，我们得到一个“巴拿赫空间”。这些空间已经被研究了大约70年。它们抓住了许多标准现象背后的基本数学，包括从一个巴拿赫空间到另一个巴拿赫空间的连续线性变换；这些线性映射的集合形成了另一个巴拿赫空间。巴拿赫空间的标准例子是由某些连续函数和可积分函数组成的空间，从而将我们的理论与微积分的理论联系起来。对这些空间的研究被称为“泛函分析”，是现代数学中一个非常重要的分支；现在，它已经成为数学荣誉学位课程最后一年的标准课程。线性空间可能有另一种结构——取两个元素的“乘积”形成一个“代数”。一个巴拿赫空间，它是一个代数并且它的乘积是连续的被称为巴拿赫代数。例如，固定巴拿赫空间上所有连续线性变换的空间本身就是一个自然的巴拿赫代数。更进一步，在某些群上的可积函数，例如实直线，形成自然“卷积”积的Banach代数。多范数空间的概念是由Dales和Polyakov提出的。它推广了有范的线性空间E，它有一个范数，通过取一系列范数，每个范数在E和它自身的n倍积空间上。这些多模必须满足某些公理。它们是一种更复杂的测量向量族“长度和形状”的方法。我们项目的第一部分是澄清和完成现有的多规范空间的描述，并提交出版。banach代数与“多banach代数”也有类似的推广。我们项目的第二部分是研究这个新概念。我们相信这些思想将对连续线性算子和各种局部紧群上的代数的研究有重要的应用。我们计划写一篇关于这些多巴拿赫代数及其应用的文章，作为回忆录提交。

A linear space is a generalization of ordinary n-dimensional space to possibly infinitely many dimensions. In such a space one can measure the length of a vector by using a `norm', so generalizing the usual concept of the length of a vector in ordinary n-dimensional space . This leads to the axiomatic definition of a `normed linear space'; if our space has an additional `completeness' property, we obtain a `Banach space'.These spaces have been studied for about 70 years. They capture the fundamental mathematics behind many standard phenomena, including those of continuous linear transformations from one Banach space to another; the collection of these linear maps forms another Banach space. Standard examples of Banach spaces are those consisting of certain continuous functions and of spaces of functions that can be integrated, thus tying our theory with that of the calculus.The study of these spaces is called `functional analysis', a very significant strand in modern mathematics; it is now a standard course in, say, the final year of honours degree programmes in mathematics.A linear space may have another structure - that of taking the `product' of two elements to form an `algebra'. A Banach space which is an algebra and which is such that the product is continuous is called a `Banach algebra'. For example the space of all continuous linear transformations on a fixed Banach space is itself a Banach algebra in a natural way. Further the integrable functions on certain groups, such as the real line, form a Banach algebra for a natural `convolution' product.The notion of a multi-norm space was introduced by Dales and Polyakov. It generalizes that of a normed linear space E, which has one norm, by a taking a sequence of norms, one on each of the n-fold product spaces of E with itself. These multi-norms must satisfy certain axioms. They are a more sophisticated way of measuring the `length and shape' of a family of vectors.The first part of our project is to clarify and complete an existing account of multi-norm spaces, and submit it for publication.There is a similar generalization of a Banch algebra to a `multi-Banach algebra'. The second part of our project is to study this new notion. We believe that these ideas wil have significan applications to the study of continuous linear operators and of algebras on various locally compact groups.We plan to write an account of these multi-Banach algebras and their applications, to be submitted as a memoir.