Is the Frechet Derivative of a Bounded Linear Operator Always the Same Operator?

[LieToMe]

I understand the Frechet derivative of a bounded linear operator is a bounded linear operator if the Frechet derivative exists, but is the result always the same exact linear operator you started with? Or, is it just "a" bounded linear operator that may or may not be known in the most general case?

 

[mathwonk]

Since the derivative is the unique best (bounded) linear approximation to the given function, the derivative of a (bounded) linear function is itself. One can check this directly from the definition of the frechet derivative given e.g. on wikipedia. I.e. A is the given operator and B is the derivative, then A-B must be a "little oh" function, or one that goes to zero at h faster than h does, i.e. one must have the limit as h-->0, of ||A(h)-B(h)||/||h|| = 0. Since certainly this holds for A=B, the uniqueness of the derivative settles it.

 

[LieToMe]

Okay, so if I start with a bounded linear operator and F-differentiate it, then I always get the same operator back, thanks.