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Section 3 Banach Manifolds

DOI: 10.1016/s0304-0208(08)70429-8
  • Mathematics


Publisher Summary This chapter discusses the different aspects of the Banach spaces. Analytic mappings are not only studied on open subsets of Banach spaces but on more general global objects called “manifolds.” In 1-dimensional complex analysis, this approach leads to the concept of Riemann surfaces. In the case of infinite dimension, many important geometric and algebraic objects are Banach manifolds in a natural way. Many basic results about analytic mappings on Banach spaces can be generalized to Banach manifolds. An example is the principle of analytic continuation. Every Banach manifold is locally and simply connected. Every connected Banach manifold admits a universal covering and simply connected space. In the theory of Riemann surfaces, the so-called fractional linear transformations or Moebius transformations play a fundamental role. The Banach manifolds studied admit fractional linear transformations as bianalytic automorphisms. For every Banach space, the set of all split subspaces of Banach space can be endowed with the structure of a Banach manifold over the center called the “Grassmann manifold.”

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