A sequence $(a_n)_{n\geq0}$ is periodic if there exists an integer $t\geq1 $ such that $a_{n+t}=a_n$ for all $n \geq0$. Is the set of periodic sequences a vector subspace of the space of sequences ?
Please kindly tell a hint for me or something recommend for me. Thank in advance !
You ask for a hint...
Check each condition for a vector space: Is the zero sequence a periodic sequence and does it act as the identity for the addition operation? Is the scalar multiple of a periodic sequence a periodic sequence, is the sum of two periodic sequences a periodic sequence? and so on ...
For example:
$(0)_{i=1}^\infty$ is the zero sequence. It is periodic with period $1$ (meaning, "with $t = 1$" in the definition you recite). This is the zero in the vector space of sequences and is the zero in the space of periodic sequences.
Let $a$ be a scalar and $b = (b_i)_{i=1}^\infty$ be a periodic sequence with period $t$, so that $b_{i+t} = b_i$ for all $i \geq 1$. Then $ab = (ab_i)_{i=1}^\infty$ is a periodic sequence with period $t$ because $ab_{i+t} = ab_i$ for all $i \geq 1$. [ Here, we are using the equality $b_{i+t} = b_i$ to replace $b_{i+t}$ with $b_i$, obtaining that $(ab_i)_{i}$ is also periodic with period $t$. ]