Determinant: Alternative Definitions Reference
Foundation for: Determinant: Continuity
Problem
Given a vector space $V$.
Consider an endomorphism $T:V\to V$.
The rank of an endomorphism:
$$\mathrm{rank}T:=\dim\left(\mathrm{im}T\right)$$
The determinant of an endomorphism:
$$\det T:=\text{???}$$
What would be a nice definition not relying on representations by matrices?
(I assume basic knowledge of Differential Geometry and Functional Analysis.)

Let $V$ be a finite-dimensional linear space over a field $\mathbb{F}$. Let $\mathcal{L}(V)$ denote the linear operators from $V$ into itself. Then I would expect that the determinant to be the unique multiplicative function $\tau : \mathcal{L}(V)\rightarrow\mathbb{F}$ for which $\tau(I)=1$. That is, $\tau(AB)=\tau(A)\tau(B)$. Is it?
