Second order sufficient condition for infinite-dimensional extremal problems

For the normed spaces X, Y with preset mapping F: X→Y and function fI: X→R (i=1,2,...k, k is given; R - the space of real numbers) the minimization problem is considered with constraints: f0(x)→MIN, F(x)=0, fj(x)≤0. Under the certain assumptions on smoothness in terms of Lagrangian formalism the second order sufficient conditions for the local minimum are obtained without a priori assumptions of finite-dimensionality and even fullness for spaces X and Y.