What is the difference between a function and a functional?

Question

When we read Functional Analysis it is said that it is a study of Functionals. I want to know how is it different from the study of functions.

Answer 1

In general a functional is a function from the linear space to the field of scalars.

In functional analysis however the concept that is most used is that of a Linear Functional. It is a "linear function" from a vector space to the field of scalars. For example $f:\mathbb{R}^2 \longrightarrow \mathbb{R}$ given by $f(\mathbf{u})=\mathbf{u} \cdot \mathbf{a}$, where $\mathbf{a}$ is some fixed vector in $\mathbb{R}^2$ is a linear functional. It is an interesting exercise (also a well-known theorem) to show that all linear functionals on $\mathbb{R}^n$ will be of this form.