Suppose that \(\displaystyle{Y}\) is not dense on \(\displaystyle{\left(X,\cdot\right)}\) . Then,
the set \(\displaystyle{\overline{Y}\neq X}\) is a closed subspace of \(\displaystyle{\left(X,\cdot\right)}\) and
according to \(\displaystyle{\rm{HahnBanach}}\)  theorem, there exists \(\displaystyle{f\in X^{\star}}\) such that
\(\displaystyle{f_{\overline{Y}}=\mathbb{O}}\) and \(\displaystyle{f=1}\) . But now,
\(\displaystyle{y\in Y\implies y\in \overline{Y}\implies f(y)=0\implies f_{Y}=\mathbb{O}}\) and according to the
hypothesis, \(\displaystyle{f=\mathbb{O}}\), a contradiction, since \(\displaystyle{f=1}\) .
