My question is:
Is the vector space containing all periodic complex sequences a finite-dimensional vector space?
My question is:
Is the vector space containing all periodic complex sequences a finite-dimensional vector space?
Davide has pretty much answered this in the comments, but here goes anyway.
Consider the sequences
$s_2=1,0,1,0,1,0,1,0,\dots$
$s_3=1,0,0,1,0,0,1,0,0,\dots$
$s_5=1,0,0,0,0,1,0,0,0,0,1,0,0,0,0\dots$
$s_7=1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,\dots$
etc, where the subscripts (and the periods) are the primes. Can you convince yourself that they are linearly independent?
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.