Corollary 3

Suppose that $K =\mathbf{C}$. Let $u$ be a positive compact endomorphism of E. The endomorphism $u$ is of finite trace if and only if the decreasing family of its eigenvalues $(\lambda_n(u))_{n\in I_E}$ is summable; the trace of $u$ is then the sum of this family.

According to th. 1 of IV, p. 149, this follows from the preceding proposition and from the definition of the sequence $(\lambda_n(u))_{n\in I_E}$ (No$^o3$ of IV, p. 151), taking account of formula (28) of EVT, V, p. 49.