continuous linear map, operator and functional

Question

Could anyone give a way to distinguish these 3 concepts? I found the following link somewhat help. https://www.physicsforums.com/threads/linear-operator-linear-functional-difference.787356/

However I need a way to distinguish them from definitions.

Answer 1

The 3 concepts you mention are closely related. All three of them are maps between (topological) vector spaces.

A linear map between vector spaces $V$, $W$ is a map $T: V \rightarrow W$ such that $T(x + y) = T(x) + T(y)$ for any vectors $x, y$ in $V$ and $T(\lambda x) = \lambda T(x)$ for any vector $x$ in $V$ and scalar $\lambda$. In other words, it's a map from $V$ to $W$ that respects our vector space operations. It takes addition in $V$ into addition in $W$, and it takes scalar multiplication in $V$ into scalar multiplication in $W$. A continuous linear map is a map that is linear in the sense described above but is also continuous if we think of $V$ and $W$ as just topological spaces. For all practical purposes, you don't have to worry about the word 'continuous' unless the spaces $V$ and $W$ are infinite-dimensional. For finite-dimensional normed vector spaces like $\mathbb{R}^n$, all linear maps are automatically continuous. So some people will use the words "linear map" and "continuous linear map" interchangeably. Note that these maps take in a vector from $V$ and spit out a vector from $W$.

An (everywhere-defined) linear operator is a continuous linear map from a vector space to itself. In other words, it's the same thing as above, but we have $T: V \rightarrow V$. (Warning: some authors use 'operator' to mean 'continuous linear map,' without the restriction that it's a map from $V$ to $V$. But most of the time 'operator' is reserved for maps with the same domain and range.) These maps take in vectors and spit out a vector from the same space.

Lastly, a linear functional (or just functional) is a continuous linear map from $V$ into the scalars (usually $\mathbb{R}$ or $\mathbb{C}$). These maps take in a vector and spit out a number. The collection of all of the linear functionals on $V$ is called the dual space of $V$ and is denoted $V^*$.

So to summarize, linear operators and linear functionals are both just special kinds of continuous linear maps (either from a space to itself or from a space to the scalars, respectively).

If you only care about the vector spaces $\mathbb{R}^n$, then the situation becomes especially nice, because it turns out that all linear maps can be rewritten as multiplication by some matrix, and all linear functionals can be rewritten as dot products against some vector. That's why we focus so much on dot products and matrix multiplication in these classes.