Is the size of the set created by the Kleene star always infinity? What about its closure?

Question

My notes state

$L^+$ denotes $LL^*$and is the closure of L under concatenation. That is, it is the smallest language that includes L and all strings that are concatenations of strings in L

Isn't the size of a Kleene star always infinity (except when the language is the empty set). For example L={a} then $L^*=\{a,aa,aaa,aaaa...\}$ therefore $|L^*|=\infty$ therefore $|LL^*|=\infty$

Is my reasoning correct?

Answer 1

No. If $L=\{1\}$, where $1$ denotes the empty word, then $L^+ = L$ and hence is finite. As you observed, if $L$ is empty, then $L^+$ is empty. In all other cases, $L^+$ is infinite.