Definition:Bounded Linear Transformation
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Definition
Let $H, K$ be Hilbert spaces.
Let $A: H \to K$ be a linear transformation.
Then $A$ is a bounded linear transformation if and only if
- $\exists c > 0: \forall h \in H: \left\Vert{A h}\right\Vert_K \le c \left\Vert{h}\right\Vert_H$
Bounded Linear Operator
Let $H$ be a Hilbert space.
Let $A: H \to H$ be a linear operator.
$A$ is a bounded linear operator if and only if
- $\exists c > 0: \forall h \in H: \left\Vert{A h}\right\Vert_H \le c \left\Vert{h}\right\Vert_H$
That is, a bounded linear operator is a bounded linear transformation from a Hilbert space to itself.
Also see
- Definition:Norm on Bounded Linear Transformation, an important concept for a bounded linear transformation
- Definition:Space of Bounded Linear Transformations
- Continuity of Linear Transformations: a linear transformation between Hilbert spaces is bounded if and only if it is continuous.
Sources
- 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next) $\S II.1$