 Solution

The bounded linear functional from part (i) satisfies $|Tx|\le \Vert T\Vert \Vert x{\Vert }_X$, where $\Vert T\Vert$ is its operator norm. HenceThese are equivalent norms. Applied to $x_n-x_m$, the inequalities show that a sequence is a Cauchy sequence for either norm exactly when it is one for the other. Applied to $x_n-x$, they show that the two norms give the same limits in $X$. Thus every Cauchy sequence has a limit in one norm precisely when it does in the other: