Gateaux and Frechet derivatives on vector valued functions

If $f:\mathbb{R}\to\mathbb{X}$ is a function from the real numbers to any normed vector space (finite or infinite dimension), and $f$ is Gateaux differentiable, is $f$ necessarily Frechet differentiable?

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Yes. In short, what differentiates (pardon the pun) Gateaux and Frechet is that derivatives in Frechet converge uniformly in the direction in the domain, while Gateaux asks only that the directional derivatives converge. Since there is only one `direction’ in the domain $\mathbb R$, these notions coincide in your case.