today my teacher ask me: but I do not how is possible it
What is the difference or relation between the following two spaces:
$$\{ u \in L^2 \ :\ \exists g\in L^2 \mbox{ such that }\int u\phi' = \int g\phi \mbox{ for all } \phi \in C_c^1 \}$$ and $$\{ u \in L^2 \ :\ \mid \mid u \mid \mid_{L^2} + \mid \mid u' \mid \mid_{L^2}<\infty \} ?$$
is it true? how is possible?
In the following I'll refer to your first space as X and to the second as Y.
As the question is set in Lebesgue-Spaces $L^2$, $u'$ means the weak derivative of $u$ which is defined by $u'\in L^2$ and $$ \int u(t)\phi'(t) dt = - \int u'(t)\phi(t)dt$$ for all $\phi\in C_c^1$. I'll just assume, that you forgot the '-' in your equation.
So let $u$ be in X. Therefore there exists a weak derivative $u'\in L^2$. We get $||u'||_{L^2}<\infty$ and obviously already have $||u||_{L^2}<\infty$ because of $u\in L^2$. Hence $u$ is in Y.
Now let $u$ be in Y. This yields the existence of a weak derivative $u'$ of $u$ which is in $L^2$. In consequence in fulfills the equation and we get $u\in X$.
Consequently, $X=Y$ and the claim is proven.