Özet:

In this article we introduce the Gateaux differential and Frechet differential in Γ-Hilbert space. We show the examples and related theorems in this space. We have noticed that two differentials mentioned above will be equal for certain condition. Also, we discuss the relative extremum and the stationary point of a functional in Γ-Hilbert space. We already investigated the characteristics of both bounded and unbounded operators of Γ-Hilbert space. Now, by using previous concept we elaborate optimization problems and extremum of quadratic functionals in Γ-Hilbert space. Here we observe that how the function of the solution of a operator equation minimizes the quadratic functionals. Finally we describe the Minimization of quadratic functionals and its related theorem via Γ-Hilbert space.

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