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As far as I can tell, which doesnt have to mean much, Frechet spaces are basicly the "closest possible" generalisation of Banach spaces, since every Frechet space is a sequential limit of Banach spaces in the category of locally convex topological vector spaces and vice versa. So that kind of makes me personally think that they should behave roughly similar...but it seems they really don't, at least when looking at calculus on them. For example when looking at derivatives we lose unique solvability of differential equations completely, we don't have an inverse function theorem and probably alot more that I am not aware of.

Is there an abstract, maybe even handwavy, reason why we "lose" these fundamental calculus tools? It seems really counterintuitive for me

Martin Brandenburg
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Benjamin Rogoll
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    I don’t know about you, but for me it is usually something remarkable if some property that holds for every element of a sequence also holds for the limit. For example, you might like to consider the fact that any countable set is the union of an increasing sequence of finite sets… or the fact that the real line is the union of an increasing sequence of compact intervals… or so on. – Zhen Lin Oct 12 '24 at 13:35
  • There still exist variants of the inverse function theorem which hold in Fréchet spaces. Picard's theorem for ODE's also has an analog for Fréchet spaces. I don't know much about Fréchet spaces, but my guess would be that we get some of the complications from the fact that the metric no longer needs to be compatible with multiplication by a scalar. – Severin Schraven Oct 12 '24 at 14:20

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