Celebratio Mathematica — Gleason

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A. M. Gleason and H. Whitney: “The extension of linear functionals defined on \( H^{\infty} \),” Pacific J. Math. 12 : 1 (1962), pp. 163–182. Dedicated to Marston Morse. MR 0142013 Zbl 0191.15202 article

We consider the Banach space \( L^{\infty} \) of (classes of) bounded measurable complex functions on the unit circle \( \Gamma_1 \). It has a subspace \( H = H^{\infty} \) consisting of those functions \( h \) which are the boundary value functions (existing almost everywhere by Fatou’s theorem) of bounded analytic functions \( \hat{h} \) in the interior \( S_1 \) of \( \Gamma_1 \). By the Hahn–Banach theorem, any bounded linear functional \( \varphi \) defined on \( H^{\infty} \) can be extended over \( L^{\infty} \) with no increase in norm. It is the primary purpose of this paper to prove that this extension is unique, provided that \( \varphi \) is defined (over \( H \)) by an integral with kernel in \( L^1 \). Without this hypothesis, uniqueness may fail.

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