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Let $g(t)$ be a convex positive function. I'm trying to show that the set $\{ |t|^ne^{\frac{g(t)}{t^n}}\}_{n\in \mathbb{N}}$ is linearly independent in the space of measurable functions on $\mathbb{R}$ (ie: $L^0(\mathbb{R})$).

I tried working in a weighted $L^2_w(\mathbb{R})$ space but these are not orthogonal so I can't use that type of argument. How could I go about showing this?

In a nutshel I'm trying to show it's a Schauder basis.

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  • $\begingroup$ If you were considering functions defined just on the positive half of the real line, then your functions for $n=0$ and for $n=1$ could be linearly dependent; in fact, they could be equal if $g(t)=(t\cdot\ln t)/(t-1)$ (if I did the arithmetic right). This particular example doesn't work for negative $t$, but I wouldn't be surprised if more complicated dependences can occur on all of $\mathbb R$ for some maliciously designed functions $g(t)$. $\endgroup$
    – Andreas Blass
    Commented Jul 22, 2016 at 4:12
  • $\begingroup$ That dounction isn't convex though, however I'm most interested in the case where $g(t)$ is constant anyway. $\endgroup$
    – ABIM
    Commented Jul 22, 2016 at 5:41
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    $\begingroup$ If g is a constant, then those functions are linearly independent just because each is infinitesimal wrto the next. $\endgroup$
    – Pietro Majer
    Commented Jul 22, 2016 at 9:36
  • $\begingroup$ Now that you've added "convex" to the question, a positive answer seems reasonably likely. $\endgroup$
    – Andreas Blass
    Commented Jul 22, 2016 at 15:30
  • $\begingroup$ So what would that be then? $\endgroup$
    – ABIM
    Commented Aug 1, 2016 at 16:59

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