So the rank r is the largest integer k for which Fk is not identically zero on E. In this case, as a non-vanishing polynomial, Fr is non-zero on an open dense subset of E. the regular elements. Any other a is a limit of regular elements a(n). Since the operator norm of L(x) gives an equivalent norm on E, a standard compactness argument shows that, passing to a subsequence if necessary, the spectral idempotents of the a(n) and their corresponding eigenvalues are convergent. The limit of Jordan frames is a Jordan frame, since a limit of non-zero idempotents yields a non-zero idempotent by continuity of the operator norm. It follows that every Jordan frame is made up of r minimal idempotents.
