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prove uniqueness of the frechet derivative.

prove uniqueness of the frechet derivative.
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Answer #1

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∥?∥≤δ⁡ε⋅∥?∥≤ε⋅δ

thus ∥A-B∥≤ε as we wanted to show.

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