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This question was asked incorrectly originally in such a way that it probably made no sense. Fixed version:

I know that $L_{p}[0,1]$ has $\ell_{2}$ as a complemented subspace, and I'm wondering if this can be generalized to include $\ell_{q}$ for other values of $q\neq 2$.

However, the only natural way I can think of to embed the space is by taking coefficients of sequences in $\ell_{p}$ and using them as coefficients of the standard Haar Schauder basis of $L_{p}$. To guarantee that the map is well-defined, I can use the usual trick of multiply by increasing powers of $2^{-1}$.

To clarify the map: For $x\in\ell_{p}$, take $T(x) = \sum\limits_{n=1}^{\infty}2^{-n}x_{n}f_{n}$, where $f_{n}$ is taken to be any normalized Schauder basis of $L_{p}[0,1]$.

I'm sure that this can't be correct though, as any textbook I've read usually devotes heavy theorem crunching (including Khintchine's inequality) just to verify the initial comment.

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See here (In particular, Theorem 4.56 on page 215.) –  David Mitra Apr 13 '12 at 3:50
    
Thanks very much. I will take a look at this. I now realize that I am skipping over the important additional requirement of being complemented in my argument, which is why it at first seemed to prove things that simply weren't true. –  Kevin Apr 13 '12 at 3:59
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This is also Theorem 6.4.21 in Albiac and Kalton's Topics in Banach Space Theory. (For $1<p,q<\infty$, $\ell_q$ embeds complementably in $L_p$ if and only if $q=p$ or $q=2$). –  David Mitra Apr 13 '12 at 4:03
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This may be of interest. From Albiac/Kalton, Theorem 6.4.19: 1) For $1\le p\le 2$, $\ell_q$ embeds in $L_p$ iff $p\le q\le 2$. 2) For $2<p<\infty$, $\ell_q$ embeds in $L_p$ iff $q=2$ or $q=p$. Moreover, if $\ell_q$ embeds in $L_p$ then it embeds isometrically. –  David Mitra Apr 13 '12 at 4:14

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