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O ° Ilw(xo,h)11 = Ilhll . 3) In this case the expression Ah is called the Frechet differential of F at Xo and is denoted by Ah = dF(xo, h). Clearly, an operator which is Frechet differentiable at Xo is continuous at this point and also Gateaux differentiable. Generally speaking, the converse is not true. 2. 2 ----+ ffi. be the operator defined by the equalities { F(xl, X2) = 1 if X2 = xI, Xl F(Xl,X2) = 0 otherwise, -I- ° 37 Derivative and differential This operator is not continuous at (0,0) and therefore is not Fhkhet differentiable.

8) holds. Thus, the number IIHII is called the norm of the operator H. D). D) is a linear normed space relatively to the norm IIHII. §7. Higher order derivatives A way of defining the derivatives and differentials of higher orders is the following. 1. D be an operator and suppose that the first variation of(x, hi) exists in a neighborhood of a point x, for any element hi EX. 1) exists in the space Z), then it is called the second variation of the operator F and is denoted by 82F(x, hI, h2)' The n-th variations are defined by induction.

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