The domain of the generator of a strongly continuous semi-group is dense

Question

Let $A$ with domain $D(A)\subset X$, ($X$ Banach) be the generator of a strongly continuous semi-group $(S(t))_{t\geq 0}$. Then $D(A)$ is dense in X.

I am not sure if this proof is correct.

I know that $\forall x\in X$, $\int_0^t S(s)xds\in D(A)$ and since the map $t\to S(t)x$ is continuous $\forall t\geq 0$ we have that:

$$\lim_{t\to 0}\int_0^t S(s)xds=x.$$

Now we need to find for any $x\in X$ a sequence $(y_n)_n\in D(A)$ such that $||y_n-x||_X\to 0$.

Define $y_n:=\int_0^{1/n}S(s)xds$, then $y_n\to x$ for $n\to \infty$ $\Longrightarrow ||y_n-x||_X\to0$, which gives the result.

Answer 1

The statement

$$\lim_{t \to 0} \int_0^t S(s) x \, ds =x$$

is not correct. Instead it should read

$$\lim_{t \to 0} \color{red}{\frac{1}{t}}\int_0^t S(s) x \, ds = x. \tag{1}$$

If we set

$$y_n := n \int_0^{1/n} S(s) x \, ds \in D(A),$$

then, by $(1)$, $y_n \to x$ as $n \to \infty$. This shows that $D(A)$ is dense in $X$.