On finite-dimensional vector spaces, we all have a reasonable idea of which functions are likely to be $C^1$ or smooth. When it comes to differentiation on Banach spaces, I find that my `intuition' absolutely fails me and is often completely wrong. Does anyone have an intuitive picture of what it means for a function between Banach spaces to be smooth (or even $C^1$), even just in simple cases like $L^p(\mathbb{R}^n)$ or $W^{1,p}(\mathbb{R}^n)$?

Most of the references I've looked at (Lang, Palais) seem to take a very formal perspective. Does anyone know of a book/paper that emphasizes a more geometric picture?

Motivation: I'm trying to understand Banach manifolds of mappings and smooth maps between them for the construction of moduli spaces in differential geometry. I thought it would be best first to understand what a smooth map between Banach manifolds actually `looks like'.

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    $\begingroup$ What is wrong with the intuition that, near any given point, a smooth function can be well approximated by its tangent? $\endgroup$ – Igor Khavkine Jun 17 '16 at 7:19
  • $\begingroup$ @Igor Khavkine Of course -- but I am asking what sort of functions actually satisfy this condition, which is certainly not something I have intuition for in infinite dimensions. But perhaps I am asking something impossible. $\endgroup$ – Mark J Jun 17 '16 at 7:33
  • $\begingroup$ Examples: constants, bounded linear functions, polynomials with bounded coefficients. But, in the end, it sounds like you are asking a highly subjective question, and it's not clear how anyone would know what a "right answer" is, other than yourself perhaps. $\endgroup$ – Igor Khavkine Jun 17 '16 at 9:00
  • $\begingroup$ Yes, those are indeed the trivial examples. You are also certainly right in observing that there is no well-defined right answer to an 'intuition' question. I'd just like to know whether anyone does happen to have an intuitive picture. At least a request for a reference that treats the subject geometrically and gives non-trivial/interesting examples is fairly well-defined. $\endgroup$ – Mark J Jun 17 '16 at 9:19

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