| EPSRC Reference: |
EP/H019405/1 |
| Title: |
Multi-norms and multi-Banach algebras |
| Principal Investigator: |
Dales, Professor HG |
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| Department: |
Pure Mathematics |
| Organisation: |
University of Leeds |
| Scheme: |
Mathematical Sciences Small Gr |
| Starts: |
01 December 2009 |
Ends: |
28 February 2011 |
Value (£): |
16,186
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| No relevance to Underpinning Sectors |
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Summary on Grant Application Form |
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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.
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.leeds.ac.uk |