Orlicz–Pettis theorem

A theorem in functional analysis concerning convergent series (Orlicz) or, equivalently, countable additivity of measures (Pettis) with values in abstract spaces.

Let ${\displaystyle X}$ be a Hausdorff locally convex topological vector space with dual ${\displaystyle X^{*}}$. A series ${\displaystyle \sum _{n=1}^{\infty }~x_{n}}$ is subseries convergent (in ${\displaystyle X}$), if all its subseries ${\displaystyle \sum _{k=1}^{\infty }~x_{n_{k}}}$ are convergent. The theorem says that, equivalently,

• (i) If a series ${\displaystyle \sum _{n=1}^{\infty }~x_{n}}$ is weakly subseries convergent in ${\displaystyle X}$ (i.e., is subseries convergent in ${\displaystyle X}$ with respect to its weak topology ${\displaystyle \sigma (X,X^{*})}$), then it is (subseries) convergent; or
• (ii) Let ${\displaystyle \mathbf {A} }$ be a ${\displaystyle \sigma }$-algebra of sets and let ${\displaystyle \mu$ :\mathbf {A} \to X} be an additive set function. If ${\displaystyle \mu }$ is weakly countably additive, then it is countably additive (in the original topology of the space ${\displaystyle X}$).

The history of the origins of the theorem is somewhat complicated. In numerous papers and books there are misquotations or/and misconceptions concerning the result. Assuming that ${\displaystyle X}$ is weakly sequentially complete Banach space, W. Orlicz[1] proved the following

Theorem. If a series ${\displaystyle \sum _{n=1}^{\infty }~x_{n}}$ is weakly unconditionally Cauchy, i.e., ${\displaystyle \sum _{n=1}^{\infty }|x^{*}(x_{n})|<\infty }$ for each linear functional ${\displaystyle x^{*}\in X^{*}}$, then the series is (norm) convergent in ${\displaystyle X}$.

After the paper was published, Orlicz realized that in the proof of the theorem the weak sequential completeness of ${\displaystyle X}$ was only used to guarantee the existence of the weak limits of the considered series. Consequently, assuming the existence of those limits, which amounts to the assumption of the weak subseries convergence of the series, the same proof shows that the series in norm convergent. In other words, the version (i) of the Orlicz–Pettis theorem holds. The theorem in this form, openly credited to Orlicz, appeared in Banach's monograph[2] in the last chapter Remarques in which no proofs were provided. Pettis directly referred to Orlicz's theorem in Banach's book. Needing the result in order to show the coincidence of the weak and strong measures, he provided a proof.[3] Also Dunford gave a proof[4] (with a remark that it is similar to the original proof of Orlicz).

A more thorough discussion of the origins of the Orlicz–Pettis theorem and, in particular, of the paper[5] can be found in.[6] See also footnote 5 on p. 839 of[7] and the comments at the end of Section 2.4 of the 2nd edition of the quoted book by Albiac and Kalton. Though in Polish, there is also an adequate comment on page 284 of the quoted monograph of Alexiewicz, Orlicz’s first PhD-student,[8] still in the occupied Lwów.

In[9] Grothendieck proved a theorem, whose special case is the Orlicz–Pettis theorem in locally convex spaces. Later, a more direct proofs of the form (i) of the theorem in the locally convex case were provided by McArthur and Robertson.[10][11]