I am trying to understand if there is an explicit way to differentiate in a Banach space. I am not a mathematician, so don't be harsh!
(In finite dimensional theory, this is easy; $Df:\mathbb{R}^d\rightarrow \mathbb{R}^d$ is just the Jacobian of $f$.)
Say I have the Banach space $C^1([0,1],\mathbb{R})$ and a countable basis such that $u = \Sigma \kappa ^iX_i $.
I want the derivative of $\gamma:\mathbb{R}\rightarrow C^1([0,1],\mathbb{R})$, say at $0$. Is it correct to assume:
$\gamma(t) = \Sigma \kappa^i(t)X_i$ $\Rightarrow$ $\dot\gamma(0) = \Sigma \dot{\kappa}^i(0)X_i$ ??
What if the map is $\gamma : C^1([0,1],\mathbb{R}) \rightarrow C^1([0,1],\mathbb{R})$ ?
Thanks for your time!