Extending L′Hôpital′s Theorem to B-Modules

An abstract version of L′Hôpital′s theorem concerning the "ratio" f(x)(g(x))−1 is proved, where f: [a, b] − X, g: [a, b] → 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) [formula] 0, g(x) [formula] 0, as x → α ∈ [a, b] is considered, when f′(x)(g′(x))−1 has a finite limit. This result complements and generalizes previous work on real- and complex-valued functions to finite- as well as to infinite-dimensional ranges. An application is also given to abstract differential equations.