Definition:Bounded Linear Operator/Normed Vector Space
< Definition:Bounded Linear Operator(Redirected from Definition:Bounded Linear Operator on Normed Vector Space)
Jump to navigation
Jump to search
Definition
Let $\struct {V, \norm \cdot}$ be a normed vector space.
Let $A : V \to V$ be a linear operator.
We say that $A$ is a bounded linear operator if and only if:
- there exists $c > 0$ such that $\norm {A v} \le c \norm v$ for each $v \in V$.
That is, a bounded linear operator on a normed vector space is a bounded linear transformation from the space to itself.