TITLE: Dimension of a vector space
QUESTION [6 upvotes]: My question is:
Is the vector space containing all periodic complex sequences a finite-dimensional vector space?

REPLY [2 votes]: 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.