Let $E,F$ be Banach spaces and $U \subseteq F$ be an open set. Let $f : U \longrightarrow F$ be a Gateaux differentiable function then $$\|f(x) - f(y)\|_F \leq \|x - y\|_E \sup\limits_{0 \leq \theta \leq 1} \left \|Df(\theta x + (1 - \theta) y) \right \|$$ whenever $[x,y]: = \{tx + (1-t)y\ |\ t \in [0,1] \},$ the line joining connecting $x$ and $y$ lies in $U.$
At the very beginning of the proof it has been stated that by Hahn-Banach theorem we can find out $\phi \in F^*$ such that $\|\phi\| = 1$ and $$\|f(x) - f(y)\|_F = \phi (f(x) - f(y)).$$
How do I find such a $\phi\ $? Which version of Hahn-Banach theorem is used here? I also don't understand how $\phi$ is linear. Can anybody please help me in this regard?
Thanks for your time.
If $u$ is a non-zero vector in a normed linear space $X$. Let $M$ be the one-dimensional space spanned by $u$. Define $\phi (cu)=c\|u\|$. This is a continuous linear functional on $M$ and its norm is $1$. By Hahn Banach Theorem this extends to a continuous linear functional on the whole of $X$ with norm $1$. Note that $\phi (u)=\|u\|$ by definition. Now take $u=f(x)-f(y)$.