Multi-norms and multi-Banach algebras

线性空间是普通n维空间到可能无限多维的推广。在这样一个空间中，人们可以通过使用“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。这导致了“赋范线性空间”的公理化定义;如果我们的空间具有额外的“完备性”性质，我们就得到了“Banach空间”。他们捕捉了许多标准现象背后的基本数学，包括从一个Banach空间到另一个的连续线性变换;这些线性映射的集合形成了另一个Banach空间。Banach空间的标准例子是那些由某些连续函数和可积分的函数空间组成的空间，从而将我们的理论与微积分的理论联系起来。它现在是一门标准课程，一个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”。一个巴拿赫空间是一个代数，它是这样的产品是连续的被称为“巴拿赫代数”。例如，在一个固定的Banach空间上的所有连续线性变换的空间本身就是一个自然的Banach代数。此外，可积函数在某些群体，如真实的线，形成一个自然的“卷积”产品的Banach代数。多范数空间的概念是由Dales和Polyakov介绍。它推广了赋范线性空间E，它有一个范数，通过取一个范数序列，一个在E的每个n重乘积空间上。这些多重范数必须满足某些公理。他们是一个更复杂的方式来衡量的“长度和形状”的一个家庭的vectors.The第一部分，我们的项目是澄清和完成现有的帐户的多范数空间，并提交出版。有一个类似的推广的一个Banach代数的一个“多Banach代数”。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，我们计划写一个关于这些多重Banach代数及其应用的报告，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.