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A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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Gateaux differentiability

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 e …