Let $L=(l_j)_{j\in\mathbb{N}}$ be a set of countably many independent real analytic functions on $[0,2\pi]$. Here and in the following, independent means that a function cannot be written as finite linear combination of the others.

Can I always find an analytic function on $[0,2\pi]$ such that the function itself, all its derivatives and its primitives (computed as indefinite integral $\int_0^x$) are independent from L? How do I construct it?

For example, if $L=(l_j):=(x^j)_{j\in\mathbb{N}}$, then $e^x$, all its derivatives and all its primitives (in this case $e^x$ as well) are independent from L, since $e^x$ is not a polynomial. How do I approach the problem in general? Thanks.

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    $\begingroup$ No matter what $L$ is, $e^{ax}$ will work for some $a$. $\endgroup$ – Wojowu Feb 12 at 10:09
  • $\begingroup$ How does one see that in a rigorous way? $\endgroup$ – Leonardo Feb 12 at 11:20
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    $\begingroup$ We may assume $L$ contains $x^j$ for all $j$. If $e^{ax}$ or any of its integrals or derivatives was linearly dependent on $L$, then $e^{ax}$ would belong to the linear span of $L$. But the latter has countable dimension, while $e^{ax}$ form an uncountable linearly independent set. $\endgroup$ – Wojowu Feb 12 at 11:36

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