Abstract This note gives a short, direct proof of the uniform smoothness of Lp spaces with . Current proofs in the literature use the uniform convexity of the Lp spaces and the duality between convexity and smoothness. As a corollary, using duality, we obtain the uniform convexity of the Lp spaces with , which is the notoriously “difficult” range, without recourse to Clarkson’s second inequality. The method of proof also gives the power-type of the modulus of smoothness of Lp spaces with ; this is known and was stated by J. Lindenstrauss and L. Tzafriri in Classical Banach Spaces II in Chapter 1, Section e. The proof was postponed to Vol. III, but this was never written.

中文翻译：

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某些勒贝格空间均匀光滑的简短证明

摘要 本笔记给出了具有 的 Lp 空间的均匀平滑性的简短直接证明。目前文献中的证明使用了 Lp 空间的均匀凸性以及凸性和平滑性之间的对偶性。作为推论，使用对偶性，我们获得了 Lp 空间的一致凸性 ，这是众所周知的“困难”范围，而无需求助于克拉克森的第二个不等式。证明方法还给出了 Lp 空间的平滑模数的幂型，其中 ; 这是众所周知的，并由 J. Lindenstrauss 和 L. Tzafriri 在 Classical Banach Spaces II 的第 1 章第 e 节中陈述。证明被推迟到卷。III，但这从来没有写过。