An elementary invariant definition of the functions of bidegree $(p,q)$

Abstract:

The alternating $r$-linear complex-valued functions of bidegree $(p,q),p + q = r$, are usually defined on a complex vector space $V$ as the span of the elements ${g_{{i_1}}} \wedge \ldots \wedge {g_{{i_p}}} \wedge {\bar g_{{j_1}}} \wedge \ldots \wedge {\bar g_{{j_q}}}$, where $\{ {g_i}:i \in I\}$ is a basis for ${V^ \ast }$ , or by means of a representation of the exterior power of a direct sum. The former definition is not a priori invariant under coordinate changes and not easily adaptable to analysis on infinite-dimensional spaces, and the latter one rests on a rather involved abstract construction. Here it is shown how to give a new coordinate-free definition of the $(p,q)$ functions by means of a simple identity which characterizes them by their action as $r$-linear maps on $V$. It seems well adapted for analysis on infinite-dimensional spaces.