= Solution
The <bounded linear functional> from part (i) satisfies $|Tx|\leq\|T\|\|x\|_X$, where $\|T\|$ is its <operator norm>. Hence
$$
\boxed{\|x\|_X\leq\|x\|_T\leq(1+\|T\|)\|x\|_X.}
$$
These 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:
$$
\boxed{(X,\|\cdot\|_T)\text{ is complete}\iff(X,\|\cdot\|_X)\text{ is complete}.}
$$
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