Derivatives of Continuously Parameterized Operators on Hilbert Spaces I'm working through Ballentine's Quantum Mechanics - A Modern Development and I've reached a section of the book where he is looking at infinitesimal transformations and generators. To do this, he assumes a family of operators continuously parametereized by a variable $s$, $U(s)$, and expands around $s=0$ as $$U=I+\frac{dU}{ds}|_{s=0} +O^2(s).$$
I don't quite understand how these derivatives are defined. I understand that we can take limits in the Hilbert space, so my first thought is something like the derivative being defined by $$\frac{dU}{ds}|\psi\rangle = \lim_{\epsilon\to0}\left(\frac{U(s+\epsilon) - U(s-\epsilon)}{\epsilon}\right)|\psi\rangle$$
for all kets in the space. Then, if this converges with respect to the norm of the space, the derivative is defined. 
Is this the right way to think about the derivative of an operation, with higher derivatives defined similarly? From here, I would assume we can talk about convergent series of operators to define the exponential. This seems to be the purview of functional analysis. Are there any recommended sources from which I could learn more about analysis with operators? 
 A: Yes this is the right way to look at this except you need $U(s)$ instead of $U(s-\epsilon)$ in your formula.
If you have a map $s\mapsto U(s)$ into the set of bounded operators on a Hilbert (e.g., unitary operators), then you have two main approaches for defining the derivative
$$
\frac{d}{ds}U(s)=\lim_{\epsilon\rightarrow 0} \frac{1}{\epsilon}(U(s+\epsilon)-U(s))\ .
$$
This depends on the topology.
1) The operator norm topology: meaning the convergence in this limit is with respect to the operator norm. This is too restrictive for QM applications.
2) The strong operator topology: that's exactly what you said, i.e., the limit means that for every ket $|\psi\rangle$, you apply $\frac{1}{\epsilon}(U(s+\epsilon)-U(s))$
and see if this converges in the Hilbert space norm. In general it does not converge for every ket. So the derivative operator $\frac{d}{ds}U(s)$ typically is an unbounded operator which is only defined on a dense subspace of the Hilbert space. Look up Stone's Theorem. Also, a good book to go more in depth into this is
"Quantum Mechanics and Quantum Field Theory
A Mathematical Primer"
by Jon Dimock.
