A fundamental theorem in the representation theoretic approach to the theory of automorphic forms is an analogue of Frobenius reciprocity; it was proved by I. Gel’fand and I. Piatetski-Shapiro in [Usp. Mat. Nauk 14, 171-194 (1959; Zbl 0121.30601); and Trans. Mosc. Math. Soc. 12, 438-464 (1963); translation from Tr. Mosk. Mat. O.-va 12, 389-412 (1963; Zbl 0136.07301)] and reproduced in their very influential book with M. Graev [Representation theory and automorphic functions, Moskau, Nauka (1966; Zbl 0138.07201); English translation: Philadelphia, W. B. Saunders (1969; Zbl 0177.18003)] where it is simply called the “Duality Theorem”. It is crucial to understanding what is meant by the concept of an “automorphic form”. In this paper the authors give an abstract, functional-analytic version of this theorem. They then combine this with the representation theory of \(\text{SL}(2,\mathbb{R})\) in order to obtain estimates of automorphic forms with respect to various norms. Although the abstract theorem is not that difficult to prove, it is remarkably effective in yielding estimates which are difficult to obtain by more conventional methods.