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My question is:

Is the vector space containing all periodic complex sequences a finite-dimensional vector space?

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Consider $e^{(k)}$ the sequence defined by $e^{(k)}_k=1$ and $e^{(k)}_j=0$ for $k\neq j$. Now take $v_p:=\sum_{k=0}^{+\infty}e^{(kp)}$. These sequences are periodic. – Davide Giraudo Oct 9 '11 at 18:12

Davide has pretty much answered this in the comments, but here goes anyway.

Consider the sequences


etc, where the subscripts (and the periods) are the primes. Can you convince yourself that they are linearly independent?

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Alternatively, if there were a finite basis of periodic sequences, with periods $p_1$, $\dots$, $p_n$, then every sequence would be a linear combination of the elements of that basis and, in particular, would have $q=p_1\cdots p_n$ as a period.

Since there do exist periodic sequences for which $q$ is not a period, your statement follows.

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