proof:
Assume that both A and B in L(?,?) satisfy the condition for the Fréchet derivative at the point ?.
To prove that they are equal we will show that for all ε>0 the operator norm ∥A-B∥ is not greater than ε.
By the definition of limit there exists a positive δ such that for all ∥?∥≤δ
∥f(?+?)-f(?)-A?∥≤(ε/2).∥?∥ and ∥f(?+?)-f(?)-B?∥≤(ε/2)⋅∥?∥ holds.
This gives
∥(A-B)?∥ =∥(f(?+?)-f(?)-A?)-(f(?+?)-f(?)-B?)∥
≤∥f(?+?)-f(?)-A?∥+∥f(?+?)-f(?)-B?∥
<ε⋅∥?∥.
Now we have
δ⋅∥A-B∥=δ⋅sup∥?∥≤1∥(A-B)?∥=sup∥?∥≤δ∥(A-B)?∥≤sup∥?∥≤δε⋅∥?∥≤ε⋅δ
thus ∥A-B∥≤ε as we wanted to show.
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