Exercise 4, § 1

We keep the notations of the preceding exercise.

a) Let $v\in \mathscr{L}^c(E)$. The radius of convergence of the formal series det(1 $-tv$) is infinite.

b) Let $(v_n)_{n\in\mathbf{N}}$ be a sequence in $\mathscr{L}^c(E)$ converging to $v\in \mathscr{L}^c(E)$. The sequence of formal series det(1 $-tv_n$) converges to det(1 $-tv$) for the topology of simple convergence of the coefficients of formal series.

c) Let $v\in \mathscr{L}^c(E)$. For every extension L of K and every $x\in L$, we denote by det(1$-$ $xv$) the value at $x$ of the formal series det(1$-tv$), which is well defined according to a). We have det(1 $+u$) det(1 $+v$) $=$ det(1 $+u+v+u\circ v$).

d) Let J be a set and $F =\mathscr{C}_0(J)$. For all $u\in \mathscr{L}^c(E; F)$ and $v\in$ $\mathscr{L}^c(F; E)$, we have det(1 $-tu\circ v$) $=$ det(1 $-tv\circ u$) as formal series. e) Let $v\in \mathscr{L}^c(E)$. We denote by Tr($v$) the opposite of the coefficient of $t$ in det(1$-tu$). If K is of characteristic zero, then

det(1 $-tu$) $=$ exp$(-\sum$ Tr($mu^m$)$t^m)$

$m\geqslant 1$

as formal series.