In this § and the next one, K will be a p-field, commutative or not. We shall mostly discuss only left vector-spaces over K; everything will apply in an obvious way to right vector-spaces. Only vector-spaces of finite dimension will occur; it is understood that these are always provided with their “natural topology” according to corollary 1 of th. 3, Chap. I-2. By th. 3 of Chap. I–2, every subspace of such a space V is closed in V. Taking coordinates, one sees that all linear mappings of such spaces into one another are continuous ; in particular, linear forms are continuous. Similarly, every injective linear mapping of such a space V into another is an isomorphism of V onto its image. As K is not compact, no subspace of V can be compact, except {0}.




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Copyright information

© Springer-Verlag Berlin · Heidelberg 1967

Authors and Affiliations

  • André Weil
    • 1
  1. 1.The Institute for Advanced StudyPrincetonUSA

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