7.4 Operators and Functionals
405
(1) If T satisfies
T (x + y) = T x + T y (x, y ∈ D)
(7.4.1)
then T is called the additive operator.
(2) If T satisfies
T (αx) = αT (x) (x ∈ D)
(7.4.2)
then T is called the homogeneous operator.
(3) If for every n 1 and any point x n , x ∈ D (n = 1, 2, · · · ), when x n →
x, there must be T x n → T x, namely when x n − x → 0, there must be
T x n − T x → 0, the T is called the continuous operator on D.
(4) If there exists a positive number M, such that for an arbitrary x ∈ D, there is
T x Mx
then T is called the bounded operator on D. Ensure that the minimum positive
number M which ensures that the above inequality holds is called the norm of
bounded operator T , and it is denoted by T . There is clearly T x T x.
(5) If
T x = αx
(7.4.3)
then T is called the similar operator of X to X . Particularly when α = 1, T is called
the unit operator or identity operator, it is denoted by I . When α = 0, T is called
the zero operator or null operator, it is denoted by θ .
(6) If for arbitrary x 1 , x 2 ∈ D, α, β ∈ P, such that
T (αx 1 + βx 2 ) = αT x 1 + βT x 2
(7.4.4)
holds, then T is called the linear operator of D to Y . In other words, the additive
homogeneous operator is called the linear operator. For a linear operator, the continuity and boundedness are equivalent. If the range of values of the linear operator T
is a set of numbers, then T is called the linear functional, it is usually denoted by
f , g etc.