\begin{definition}[Definition:Bounded Linear Functional/Normed Vector Space]
Let $\mathbb F$ be a subfield of $\C$. 
Let $\struct {X, \norm \cdot}$ be a normed vector space over $\mathbb F$.
Let $f : X \to \mathbb F$ be a linear functional.
We say that $f$ is a '''bounded linear functional''' {{iff}}:
:there exists $C > 0$ such that $\cmod {\map f x} \le C \norm x$ for each $x \in X$.
\end{definition}