Deconstructing an “Elementary Proof”: The Case of L *_p = L_q

Chapter 11 of N. L. Carothers’ A Short Course on Banach Space Theory includes a section entitled “An elementary proof that L^*_p = L_q”. We argue that the argument, while elegant, is not elementary in any standard pedagogical sense: it relies on nontrivial facts such as uniform convexity (Clarkson’s inequalities), norm attainment and its uniqueness on the unit sphere, and differentiability statements that require careful justification. We make these dependencies explicit, point out several technical imprecisions, and provide streamlined, fully justified versions of two key steps: McShane’s lemma and the limit identity in Lemma 11.19. Our goal is twofold: (i) to clarify the logical scaffolding behind the identity L^*_p = L_q and (ii) to offer a didactic blueprint for instructors, highlighting common pitfalls and´proposing classroom-tested fixes.