Let $X$ be a normed space and let $Y$ be a dense linear subspace of $X$. Define $\phi:X^*\rightarrow Y^*$ by $\phi(f)=f|_Y$. Then I want to show $\phi$ is well-defined. And here is my attempt:
For all $f\in X^*$, $f|_Y$ is a continuous linear functional on $Y$; that is, $f|_Y\in Y^*$. If $f_1=f_2\in X^*$, then $\phi(f_1)=f_1|_Y=f_2|_Y=\phi(f_2)$. Hence, $\phi:X^*\rightarrow Y^*$ is well -defined.
But my professor said my proof was wrong in logic. It seems ok to me. Can anyone help me? Thank you!
Perhaps you should more explicitly show that $f|_Y$ is bounded:
$$\|f|_Y\|_{Y^*} = \sup_{y \in Y, \|y\| = 1} |f|_Y(y)| = \sup_{y \in Y, \|y\| = 1} |f(y)| \le \sup_{y \in X, \|y\| = 1} |f(y)| = \|f\|_{X^*}$$