Multilinear functions

  • W. H. Greub
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 136)

Abstract

Consider, for every p≧1, the linear space T p (E)of p-linear functions \( \Phi :E\underbrace {x \cdot \cdot \cdot x}_pE \to \Gamma \). For p =1, we have T1 (E)= E*. It will be convenient to extend the definition of T p (E) to the case p = 0 by setting T0 (E) Γ, The product Φ Ψ of a p-linear function Φ and a q-linear function Ψ is defined to be the (p +q)-linear function
$$ \eqalign{ & (\Phi \Psi )({x_1},...,{x_{p + q}}) = \Phi ({x_1},...,{x_p})\Psi ({x_{p + 1}}...{x_{p + q}}) \cr & \quad \quad \Phi \in {T_p}(E),\;p\underline{\underline {\; > }} \;1\quad \Psi \in {T_q}(E),\;q\;\underline{\underline > } \;1. \cr} $$
(8.1)

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Copyright information

© Springer-Verlag Berlin · Heidelberg 1967

Authors and Affiliations

  • W. H. Greub
    • 1
  1. 1.Mathematics DepartmentUniversity of TorontoTorontoCanada

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