# 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 $v \in V$.

### Topological Vector Space

Let $\GF \in \set {\R, \C}$.

Let $X$ and $Y$ be topological vector spaces over $\GF$.

Let $T : X \to Y$ be a linear transformation.

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

for each von Neumann-bounded subset $E$ of $X$, $T \sqbrk E$ is von Neumann-bounded.