Multilinear functions

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)

Throughout this chapter E denotes a vector space of finite dimension ana E* the space of linear functions in E.