(a) If $T$ is a bounded linear functional then $|T(f)|\leq \|T\|_*\|f\|$ for all $f\in X$.
(b) $\|T\|_*=\sup\{|T(f)|\mid f\in X,\,\|f\|\leq 1\}=\sup\{|T(f)|\mid f\in X,\,\|f\|=1\}$
(c) A linear function is bounded if and only if it is continuous (Hint: use (b) for the “if" part of the statement)
For a normed linear space $X$, a linear function $T$ on $X$ is said to be bounded if there is an $M≥0$ for which $|T(f)|\leq M\|f\|$ for all $f\in X$.
$\|T\|_*=\inf\{M\geq 0\mid|T(f)|\leq M\|f\|,\text{ any }f\in X\}$