# On the Definition of Gateaux Derivative

My question is about two different definitions Gateaux derivative. I have seen the following two definitions but whether they are equivalent or which one is better to use I am not sure about:

Definition 1: Derivative at $f$ with increment $h$ is defined as \begin{align*} \lim_{ \theta \to 0} \frac{F[f+\theta h]- F[f]}{\theta} \end{align*}

Definition 2: Derivative at $f$ with increment $h$ is defined as \begin{align*} \lim_{ \theta \to 0} \frac{F[(1-\theta)f+\theta h]- F[f]}{\theta} \end{align*}

Are these the same? Is there an advantage of using Def 1 vs Def 2?

Def.1 is how I been thought the Gateaux derivative over vector space.

Def.2 I seen in the context of probability theory where derivative is taken over all probability distributions. The paper I read is also calling it Gateaux derivative.

I feel that the second one use if we want to take derivative over convex sets only. That is why we use $(1-\theta)f+\theta h$. But not sure completely?

Another question is if stationary point with one definition is also a stationary point with another?

Thank you in advance for any help.

• If you replace $h$ by $h - f$ in the first definition, you arrive at the second. Hence, they define different things. The typical definition is the first one.
– gerw
Mar 6, 2015 at 21:10
• Thank you. Do you possibly know what is the second definition called? I saw it in a paper they referred to it as weak derivative. But didn't give any reference to the origin.
– Boby
Mar 6, 2015 at 21:11
• Derivative with increment $h-f$ ;)
– gerw
Mar 6, 2015 at 21:12
• @gerw Have you hear of weak derivative?
– Boby
Mar 6, 2015 at 21:17
• Yes, I know weak derivatives.
– gerw
Mar 7, 2015 at 20:36

The Gâteaux derivative has a standard definition, which is your definition 1. It may be the case that some author in a particular situation is envisaging the limit in your definition 2, which is certainly not far away from the "official" Gâteaux derivative. But there is an essential difference: The "official" derivative is linear in the variable $h$, while the expression in definition 2 is not.
We see the same discrepancy when considering Taylor expansions in ordinary calculus. The $n$-jet, i.e., the $n$-th degree Taylor polynomial, of a function $f$ at some point $a$ can be written as a polynomial in the "primary" variable $x$ as $$j_a^{n}(x)=\sum_{k=0}^n {f^{(k)}(a)\over k!}(x-a)^k\ ,$$ but equivalently as a polynomial in the increment variable $X:=x-a$: $$j_a^{n}(X)=\sum_{k=0}^n {f^{(k)}(a)\over k!}X^k\ .$$ The second view, which correspnds to your definition 1, seems much more natural.