For Banach spaces $X,Y$ and an open subset $U$ of $X$ a function $f:U\to Y$ is $C^1$ if $U\to L(X,Y)$, $x\to f'(x)$ is continuous where, by definition, the derivative $f'(x)$ is a continuous linear operator from $X$ to $Y$ satisfying $$\|f(x+h)-f(x)-f'(x)(h)\|\_Y = o(\|h\|\_X)$$ and $L(X,Y)$ is the Banach space of continuous linear operators endowed with the operator norm. Then $f'$ is again a map between Banach spaces and one can define higher derivatives.


For Frechet instead of Banach spaces there are many different topologies on $L(X,Y)$ but none of them is Frechet. To stay in the category one thus needs a different definition. In his exposition of the Nash-Moser theorem
[Hamilton](http://www.ams.org/bull/1982-07-01/S0273-0979-1982-15004-2) therefore tells $f$ continuously differentiable if all directional derivatives $Df(x)(v)=\lim\limits\_{t\to 0}(f(x+tv)-f(x))/t$ exist in $Y$ and $Df: U\times X \to Y$ is continuous. He then writes:
*Note well that this definition of a $C^1$ map does not agree with the usual
definition for a Banach space. The derivative will of course be the same map (i.e. $Df(x)(v)=f'(x)(v)$),
but our continuity requirement is weaker.*


What is an example (in Banach spaces) showing that the latter definition is *strictly* weaker than the former?



$C^1$-functions on Banach spaces
In Banach spaces, continuity of $Df:U\times X \to Y$ and linearity in the second variable implies that $Df: U\to L(X,Y)$ is continuous for the topology of uniform convergence on compact subsets of $X$, and not the operator norm in general.
If $f$ is $C^2\_c$ then $Df: U\to L(X,Y)$ is actually continuous into the operator norm.


Since I am travelling, I am now quite incomplete. Sorry.


EDIT:
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I had the opportunity to look up the literature now. An example on $\ell^2$, due to Smolyanov, is in  

12.13 of:


* Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997. [(pdf)](http://www.mat.univie.ac.at/%7Emichor/apbookh-ams.pdf)




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Edit by Jochen Wengenroth


The example is the following: Take a smooth function $\varphi$ with support in $[-1/4,1/4]$ and $\varphi(0)=1$. Define $f\_k:\ell^2\to \mathbb R$ by $x\mapsto \varphi(k(x\_k-1)) \prod\limits\_{j<k}\varphi(jx\_j)$ and $f=\sum\limits\_{k=1}^\infty \frac{1}{k^2} f\_k$.


