  This paper is devoted to establishing an enhanced Fritz John type first-order
necessary condition for a general constrained nonlinear infinite-dimensional
optimization problem. Unlike traditional constraint qualifications in
optimization theory, a condition of finite codimensionality is employed to
ensure the existence of nontrivial Lagrange multipliers. As applications,
first-order necessary conditions for optimal control problems of some
deterministic/stochastic control systems are derived in a unified manner.
Compared with the existing constraint qualifications, the finite
codimensionality condition, which is equivalent to some suitable {\it a priori}
estimates, can offer a more straightforward verification process in these
applications.
