Definition:Bounded Linear Transformation

Definition

Normed Vector Space

Let $\struct {V, \norm \cdot_V}$ and $\struct {U, \norm \cdot_U}$ be normed vector spaces.

Let $A : V \to U$ be a linear transformation.

We say that $A$ is a bounded linear transformation if and only if:

there exists $c > 0$ such that $\norm {A v}_U \le c \norm v_V$ for each $v \in V$.

Inner Product Space

Let $\struct {V, \innerprod \cdot \cdot_V}$ and $\struct {U, \innerprod \cdot \cdot_U}$ be inner product spaces.

Let $\norm \cdot_V$ and $\norm \cdot_U$ be the inner product norms of $V$ and $U$ respectively.

Let $A : V \to U$ be a linear transformation.

We say that $A$ is a bounded linear transformation if and only if:

there exists $c > 0$ such that $\norm {A v}_U \le c \norm v_V$ for each $v \in V$.

Also see

• Definition:Norm on Bounded Linear Transformation
• Definition:Space of Bounded Linear Transformations
• Definition:Bounded Linear Operator

• Continuity of Linear Transformations: a linear transformation between Hilbert spaces is bounded if and only if it is continuous.