Do the set of functions: $\{1 , \frac{1}{x+1}, \frac{1}{(x+1)(x+2)}, \dots \}$ form a (non-orthonormal) Schauder basis of $L^p([0,1])$ for $1 \le p < \infty$? How can I prove or disprove this statement?
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1What kind of basis do you mean? Do you mean a Hamel basis? – Severin Schraven Oct 21 '18 at 16:08
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@SeverinSchraven edited – Television Oct 21 '18 at 16:10
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1I can think of some reasons this seems unlikely; what are your thoughts? – SBK Oct 21 '18 at 16:29
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Just consider the characteristic function of $[0.5, 1]$ and show that you cannot approximate it with elements of your basis. – Manlio Oct 21 '18 at 16:49