  We study a singular Hermitian metric of a vector bundle. First, we prove the
sheaf of locally square integrable holomorphic sections of a vector bundle with
a singular Hermitian metric, which is a higher rank analogy of a multiplier
ideal sheaf, is coherent under some assumptions. Second, we prove a
Nadel-Nakano type vanishing theorem of a vector bundle with a singular
Hermitian metric. We do not use an approximation technique of a singular
Hermitian metric. We apply these theorems to a singular Hermitian metric
induced by holomorphic sections and a big vector bundle, and we obtain a
generalization of Griffiths' vanishing theorem. Finally, we show a
generalization of Ohsawa's vanishing theorem.
