In proving the chain rule, Duistermaat and Kolk use $$\lim_{x \to a}\psi(f(x))\phi(x) = \lim_{x \to a} \psi(f(x))\lim_{x \to a}\phi(x) = (Dg(f(a))Df(a)$$ where $\phi(x)$ and $\psi(y)$ are the linear functions from Hadamard's lemma for $f(x)$ and $g(y)$ respectively.
My question regards the first equality: $$\lim_{x \to a}\psi(f(x))\phi(x) = \lim_{x \to a} \psi(f(x))\lim_{x \to a}\phi(x)$$ Is the limit of the product of two operator valued functions the product of the limits of the functions?
The book claims it follows from the limit of the inner product of two functions is equal to the inner product of their limits, i.e. $$\lim(f_1,f_2)=(\lim f_1,\lim f_2)$$ Yet this doesn't apply here at all.
Many thanks for any help.
The result that for sequences $x_n \rightarrow x$, $y_n\rightarrow y$ $$ \lim_{n \rightarrow \infty} x_ny_n =xy$$ Along with the result that: $$ \lim_{x \rightarrow a} f(x) = c \, \text{iff} \lim_{n \rightarrow \infty} f(a_n) = c \, $$ $$\forall \{a_n\} \, \text{s.t} \,a_n \neq a \vee a_n \rightarrow a$$ This immediately implies
As we are given that the limits of $\psi, \phi$ exist, the result follows.