Bounded Linear Functionals and the Dual Space of a Real Normed Space

In the setting of Separable Real Normed Spaces and Hahn–Banach: Notation, let $X$ be a real normed space and let $F$ be a linear functional on $X$.

(i) Bound. A real number $C$ is a bound for $F$ if

$$|F(x)|\le C\,\lVert x\rVert\qquad\text{for every }x\in X.$$

(ii) Set of bounds. The set $B_{F}$ of nonnegative bounds of $F$ is

$$B_{F}=\{M\in\mathbb{R}:M\ge0\text{ and }M\text{ is a bound for }F\}.$$

(iii) Bounded functional. $F$ is bounded if it has a bound.

(iv) Dual space. The dual space $X^{*}$ of $X$ is the set of bounded linear functionals on $X$.