Continuity of identiy mapping in the weak topology

Question

I am trying to show that the identity mapping $$id :(M,\parallel \parallel) \longrightarrow (E,\sigma(E,E^*)) $$ is continuous. such that $E$ is a normed space and $M$ a vector subspace of $E$. Using the fact that induced topology on M by the weak topology of E are the same. $$ \sigma(M,M^*) = \sigma(E,E^*)\cap M$$ What I have tried so far : if we consider $O$ an open subset in $\sigma(E,E^*) $ $$id^{-1}(O) = O \cap M$$ which is open in the weak toplogy in $M$. But this doesn't prove that $id^{-1}(O)$ is open in $M$ strongly .