product rule for linear maps Having a linear map $L(t)$ on a banach space and an element $v(t)$ of that space, both beeing dependend of a paramter $t$. Now the differentiation of the output of the map applied on v(t) is given as $ \frac{d}{dt}L(t) v(t) = L'(t) v(t) + L(t) v'(t)$ ala product rule. But if i read it as $\frac{d}{dt} L(t)(v(t)) = L'(t)(v(t)) \cdot v'(t)$ ala chain rule i would get sth different, why isn't it here applicable that way?

You can apply the chain rule, but you need to consider the bilinear function $(L, v) \mapsto L(v).$ Its derivative at $(L_0, v_0)$ is $(L, v) \mapsto L_0 v + L v_0.$ Notice also that the derivative of $t \mapsto (L(t), v(t))$ is $(L'(t), v'(t)).$ If you now make substitutions, you will see that chain rule and product rule agree.
