> *Definition 1*: a **linear functional** $f$ is a linear operator with its domain in a vector space $X$ and its range in a scalar field $F$ defined in $X$.
The norm can be a linear functional $\|\cdot\|: X \to F$ under the condition that the norm is linear. Otherwise, it would solely be a functional.
> *Definition 2*: a **bounded linear functional** $f$ is a bounded linear operator with its domain in a vector space $X$ and its range in a scalar field $F$ defined in $X$.
## Dual space
> *Definition 3*: the set of linear functionals on a vector space $X$ is defined as the **algebraic dual space** $X^*$ of $X$.
From this definition we have the following.
> *Theorem 1*: the algebraic dual space $X^*$ of a vector space $X$ is a vector space.
??? note "*Proof*:"
Will be added later.
Furthermore, a secondary type of dual space may be defined as follows.
> *Definition 4*: the set of bounded linear functionals on a normed space $X$ is defined as **dual space** $X'$.
In this case, a rather interesting property of a dual space emerges.
> *Theorem 2*: the dual space $X'$ of a normed space $(X,\|\cdot\|_X)$ is a Banach space with its norm $\|\cdot\|_{X'}$ given by
