Fréchet derivative is unique

Theorem The Fréchet derivative is unique.
Proof. Assume that both A and B in L(𝖵,𝖶) satisfy the condition for the Fréchet derivative ( at the point 𝐱. To prove that they are equal we will show that for all ε>0 the operator normMathworldPlanetmath A-B is not greater than ε. By the definition of limit there exists a positivePlanetmathPlanetmath δ such that for all 𝐡δ

f(𝐱+𝐡)-f(𝐱)-A𝐡ε2𝐡 and f(𝐱+𝐡)-f(𝐱)-B𝐡ε2𝐡

holds. This gives

(A-B)𝐡 =(f(𝐱+𝐡)-f(𝐱)-A𝐡)-(f(𝐱+𝐡)-f(𝐱)-B𝐡)

Now we have


thus A-Bε as we wanted to show.

Title Fréchet derivative is unique
Canonical name FrechetDerivativeIsUnique
Date of creation 2013-03-22 16:08:35
Last modified on 2013-03-22 16:08:35
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 12
Author Mathprof (13753)
Entry type Theorem
Classification msc 46G05
Related topic derivative