Fréchet differentiability from Gâteaux differentiability

Let $X$ be a Banach space and $\Omega \subset X$ be open.

The functional $f$ has a Gâteaux derivative $g \in X'$ at $u \in \Omega$ if, $\forall h\in X,$ $$\lim_{t \rightarrow 0}[f(u+th)-f(u)- \langle g,th \rangle]=0$$

How can I prove the following:

If $f$ has a continuous Gâteaux derivative on $\Omega$, then $f \in C^1(\Omega,\mathbb R)$.

8
2022-07-25 20:46:59
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Answers: 2

Theorem 1-2 from Saaty's "Modern Nonlinear Equations" proves this using a combination of the mean value theorem and the Hahn Banach Theorem.

3
2022-07-25 22:24:01
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My understanding of the problem: we know that the Gâteaux derivative exists and is continuous; we want to prove that it is actually the Fréchet derivative. Pick a point $x_0$. Subtract a linear functional from $f$ so that $f\,'(x_0)=0$. For any $\epsilon>0$ there is a neighborhood of $x_0$ in which $|f\,'|<\epsilon$. By the Mean Value theorem $|f(x)-f(x_0)|\le \epsilon |x-x_0|$ in this neighborhood. Hence $f\,'(x_0)=0$ in the Fréchet sense.

3
2022-07-25 22:23:46
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