# Can not find reference for "uniform convexity implies existence of unique conjugate" mentioned by Pettis

In A proof that every uniformly convex space is reflexive in footnote 3 (available at that link without a paywall), author Billy Pettis mentions that the first half of Lemma 1 in that paper "was discovered independently by J. A. Clarkson and E. J. McShane in 1936".

I attempted to find the corresponding publications, but could only find the publication Semi-continuity of integrals in the calculus of variations of E. J. McShane from 1936, which seems to concern unrelated topics (although I only skimmed the paper, but uniform convexity and dual spaces are not even mentioned). His paper on Jensen's inequality also does not contain this assertion, if I am not mistaken. I also searched Google Scholar for paper quoting the paper by Clarkson mentioned below, which introduces uniformly convex spaces, which are critical for this result, but I found none published between 1936 and 1949. But in his work Linear Functionals on Certain Banach Spaces published in 1950, the existence part of this lemma is proven in his proof of Lemma 3 (p. 403) using exactly the same arguments as in Pettis' proof. This paper also seems to be the only one of his from 1936-1950 where uniform convexity is mentioned.

The only paper by James A. Clarkson from 1936 is the famous "Uniformly convex spaces", which, introduces the uniformly convex spaces and, if I am not mistaken, also does not contain this assertion. Later papers by this author seem to, again, concern very different properties.

Could the footnote by Pettis also be understood to mean that those mathematicians discovered but did not formally publish this result?

For reference, the statement is

If the Banach space $$X$$ is uniformly convex and $$\gamma_0 \in \overline{X}$$ (the dual space of $$X$$) with $$\| \gamma_0 \| \ne 0$$, then there exists a unique $$x_0 \in X$$ such that $$\| x_0 \| = 1$$ and $$\gamma_0(x_0) = \| \gamma_0 \|$$.

I have seen the latter property being called "$$x_0$$ achieves the least upper bound (l.u.b) for $$\gamma_0$$", e.g. in E.R. Lorch: "A curvature study of convex bodies in Banach spaces", 1953. This property also appears in Fortet: Remarques sur les éspaces uniformément convexes translation on page 45.

• One thing you can try is to look over the conference talk titles and conference abstracts published in Bulletin of the American Mathematical Society for 1935-1940. Example for published conference talk titles. Example for published conference abstracts. In the mid 1990s I went through every Bull. AMS volume (library hardcopy) (continued) Mar 28, 2022 at 15:12
• and photocopied every published abstract of conceivable interest to me, and I re-did this about 10-15 years later (knowing more about more things at this later time). This is how, for example, I knew about the abstracts I cited in one of my comments to this mathoverflow question (regarding my interests, note one is a Baire-typical result). However, although I definitely recognize the names involved (and have photocopies of many papers and abstracts by them), the specific topic area for you question has never been within the scope of my interests. Mar 28, 2022 at 15:18