8. Linear Operators

Question 1

8.1 Define what it means for a map T : X → Y to be linear.

Answer 1

Definition 8.1:
If T (λx + μy) = λT x + μT y
for all x,y∈X and all λ,μ∈F

Question 2

Define what it means for a map T : X → Y to be a bounded
linear operator.

Answer 2

Definition 8.16: A map T : X → Y is a bounded linear operator if it
is linear and there exists a constant M ≥ 0 such that ‖T x‖ ≤
M ‖x‖ for all x ∈ X.

Question 3

Define the kernel KerT and image ImT of the linear operator T : X -> Y

Answer 3

Defintion 8.10
Ker T = {x ∈ X : T x = 0} and
Im T = {y ∈ Y : y = T x for some x ∈ X}