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Let $P$ be a finite dimensional contractive (norm 1) projection in $L_p$, $1 < p < \infty$. Then $P$ is of the following form:

$Pf = \sum_{k=1}^n g_k \int h_kf$

Where $\|g_k\|_p = \|h_k\|_q = \int h_kg_k = 1$, $g_k$ have disjoint supports and $g_k h_j \equiv 0$ for $j \ne k$.

This was found in a paper by Pełczyński and Rosenthal - "Localization techniques in $L_p$ spaces" without any reference and I was unable to prove it.

The purpose of this is to show that every monotone basis in $L_p$ is unconditional without using the fact that every contractive projection is (up to an isometry) a conditional expectation (or the fact that every constant preserving projection is a conditional expectation, which, after some work implies the aforementioned fact).

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    $\begingroup$ Check volume 2 of Lindenstrauss-Tzafriri "Classical Banach spaces". I think it also is in Lacey's book that has Isometric and Banach spaces in its title. $\endgroup$
    – Bill Johnson
    Commented Nov 2, 2012 at 20:31

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Contractive projections in $L_p$ were characterized by Ando (Pacif. J. Math. 17 (1966), 391-405). Discrete case was discussed in Theorem 2.a.4 of Lindenstrauss-Tzafriri, Classical Banach spaces, Volume 1. Hopefully you can derive the desired result using one of the mentioned above.

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