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Let $\{e_{n}(x)\}_{n=0}^{\infty}$ be orthonormal basis of Hilbert space $L^2(\mathbb{R})$. If $\{e_{n}(x)\}_{n=0}^{\infty} \subset L^p(\mathbb{R})$ for some $p\geq 1$, is the $\{e_{n}(x)\}_{n=0}^{\infty}$ Schauder basis for $L^p(\mathbb{R})$?

Any reference?

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  • $\begingroup$ You want $\subset$, not $\in$ $\endgroup$
    – zhw.
    Jan 3, 2018 at 18:04

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