Linear Functionals on Linear Spaces

Definition: Let $X$ be a linear space. A Linear Functional or simply, Functional on $X$ is a linear operator $\varphi : X \to \mathbb{C}$.

In other word, a linear function on a linear space $X$ is simply a linear map $\varphi$ whose range is $\mathbb{C}$. Sometimes we will restrict linear functionals on $X$ to be real-valued. We give such functionals a special name.

Definition: Let $X$ be a linear space. A Real Linear Functional or $\mathbb{R}$ Functional on $X$ is a linear operator $\varphi : X \to \mathbb{R}$.