I'm a newbie in mathematical physics and is currently reading Sadri Hassani's book entitled Mathematical Physics - A Modern Introduction to Its Foundations. I came upon this page in the book and I would like to ask for some clarifications.
It says in the definition that the derivative is defined for a mapping $H$ from the set of real numbers to the set of linear endomorphisms of $V (L(V))$. It also says that the derivative of such a mapping $H$ is an element of $L(V)$ itself. With this in mind, I would like to ask the following questions.
- Are $U$ and $T$ both mappings from the set of real numbers to $L(V)$? In this case, how is $UT$ defined? It can't be the composition of $U$ and $T$ since the range of $T$ would be $L(V)$ while the domain of $U$ is the set of real numbers.
- Similarly, how are $(\mathrm dU/\mathrm dt)T$ and $U(\mathrm dT/\mathrm dt)$ defined?
Thank you so much!
This is all to be understood pointwise, i.e., the derivative of $H$ at a fixed time $t$ is an element of $L(V)$ and the value of $UT$ at time $t$ is $U(t)T(t)$ (which is the composition of these linear operators) and the same holds for $U\frac{dT}{dt}$ and $\frac{dU}{dt}T$.