Definition:Bounded Linear Transformation
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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:
- $\forall v \in V: \exists c > 0: \norm {A v}_U \le c \norm v_V$.
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.
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.