THE ADJOINT MAP AND DUAL MAP OF MULTILINEAR MAP

Abstract

Let Vibe the vector spaces and Vi* be their dual spaces for 1 <= i <= r . In this paper, (V1 X V2 X ... X Vr)* = V* * 1 X V * 2 X ... X Vr is examined. Furthermore, the dual map F* and the adjoint map F' of the r-linear map F: V1X V2X ... X Vr-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: V1X V2X ... X Vr-W1 X W2 X ... X Wr is defined and for their matrices, the equality [F*]=[F]T is true, for the vector spaces Vi and Wi .