Functions of Vectors

A function is a map $f : V \to S$ .

Easy to imagine two useful categories of such functions: field elements or vectors. You can map vectors to arbitrary sets.

Functional

A map from a vector to a field element is defined as a functional.

A map from a vector to a vector is defined as a operator.

A map from any vector space $V$ is linear if it distributes over the vector space operations:

$V^{*}$ is the dual space of $V$ , where elements of $V^{*}$ are all linear functionals $f : V \to F$ .

Notation: Define $f (| a ⟩) \equiv ⟨ f | a ⟩ \in F$ .

The Dual Space is a Vector Space.

$F (| a ⟩) = F | a ⟩ = | F a ⟩$

$M : R^{n} \to R^{m}$ . $m \times n$ matrices.
$M : V \to V$ , with $V$ being a funcation space. Derivatives, fourier transform, Laplace, integral.

If $F = C$ and $f (α | a ⟩ + β | b ⟩) = α^{*} f (| a ⟩) + β^{*} f (| b ⟩)$ then the operator is called anti-linear.