From Euclidean to Hilbert Spaces

6Bounded Linear Operators in Hilbert Spaces

A function A : V W, with V and W normed vector spaces on the same field , is known as a linear operator between V and W if:

To simplify the notation, the parentheses may be omitted in later occurrences, writing Ax in place of A(x). V is the domain of A; the set:

is the codomain or image of A, and W is the destination set of A.

Basic examples are shown below.

1) The identity operator: id : V → V , id(x) = x ∀x ∈ V and the null operator: 0 : V → V , 0(x) = 0_V ∀x ∈ V ;

2) The differential operator: this is defined on a space of differentiable functions which may change according to the particular application we are interested in. As a concrete example, consider the first-order differential operator: . dom(D₁) = {f ∈ L²[a, b] ⋂ ¹[a, b] : f′ ∈ L²[a, b]}, where a < b are real constants, could be a perfectly valid domain for D₁. Then:

Similarly, ...