THE ADJOINT MAP AND DUAL MAP OF MULTILINEAR MAP

Abstract

Let be the vector spaces and V_i^* be their dual spaces for 1≤i≤r . In this paper, (V_1×V_2×…×V_r )^*≅V_1^*×V_2^*×…×V_r^* is examined. Furthermore, the dual map F^* and the adjoint map F^' of the r-linear map F:V_1×V_2×…×V_r→W are defined and for their matrices, the equality [F^* ]=[F^' ]=[F ̅ ]^T is found. Analog, the dual map F^* of the r-variable vector valued linear map F:V_1×V_2×…×V_r→W_1×W_2×…×W_r is defined and for their matrices, the equality [F^* ]=[F ̅ ]^T is true, for the vector spaces V_i and W_i .