Definition:Bounded Linear Operator

Definition

Normed Vector Space

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.

Inner Product Space

Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.

Let $\norm \cdot$ be the inner product norm for $V$.

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 an inner product space is a bounded linear transformation from the space to itself.