Differentiation of operators and functionals

Definition 73 A linear operator A in a Hilbert space H is called self-adjoint (or symmetric ) if 

Definition 74 A symmetric operator is said to be positive on some subset S C H if for allxG S 

Definition 75 A linear operator A in a real Hilbert space H is called positively determined in some subset S C H, if we can find a constant 7 0 such that, for all X 5, the following relationship holds  

The last definition can be extended to the case of the complex Hilbert space 

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