Definition:Bounded Linear Transformation/Normed Vector Space

Definition

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 $v \in V$.

Also see

• Definition:Bounded Linear Operator on Normed Vector Space

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $11.1$: Bounded Linear Maps