Extending l'Hôpital's theorem to \(B\)-modules (Q1323105)

An abstract version of L'Hôpital's theorem concerning the ``ratio'' \(f(x)(g(x))^{-1}\) is proved, where \(f: [a,b]\to X\), \(g: [a,b]\to A\), \(A\) being a unital Banach algebra, \(X\) a Banach module over \(A\), and \((a,b)\) a bounded or unbounded real interval. Here, the case \(f(x)@>X>>0\), \(g(x)@>A>> 0\), as \(x\to \alpha\in[a,b]\) is considered, when \(f'(x)(g'(x))^{-1}\) has a finite limit. This result complements and generalizes to finite- as well as to infinite-dimensional ranges previous work on real- and complex-valued functions. An application is also given to abstract differential equations.