Pazy's book defines semigroups as follows. Let $X$ be a Banach space. A one parameter family $T(t)$, $0< t < \infty$, of bounded linear operators from $X$ into $X$ is a semigroup of bounded linear operators on $X$ if
(i) $T(0) = I$, ($I$ is the identity operator on $X$).
(ii) $T(t + s) = T(t)T(s)$ for every $t, s \geq 0$ (the semigroup property).
What is the importance of $X$ Banach space in the theory of semigroups of linear operators? Does anything break if I require that $X$ just be a complete metric space?