I have a question about the Gateaux differentiability of a map on a particular space. To be more specific, let $f: U \to X$ be a map where both $U$ and $X$ are Hilbert spaces.
Now, assume that there exists a Banach space $X_1$ and a Hilbert space $X_2$ such that:
(I) The embedding $X_1 \hookrightarrow X_2$ is continuous.
(II) The embeddings $X \hookrightarrow X_1, X_2$ are both compacts.
(III) $f: U \to X_2$ is continuously Gateaux differentiable.
QUESTION: Can I say something about the differentiability of $f: U \to X_1$?
It would be ideal if I could prove that $f: U \to X_1$ is also continuously Gateaux differentiable. I thought one could be able to explore the chain $$f: U \to X \hookrightarrow X_1 \hookrightarrow X_2$$ and use some sort of interpolation ideas.
If you have seen anything similar to that in some papers or books, I am happy to take a look at them.
I really appreciate any help you can provide.
