Definition:Norm/Bounded Linear Transformation/Definition 2
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Definition
Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.
Let $A: X \to Y$ be a bounded linear transformation.
The norm of $A$ is the real number defined and denoted as:
- $\norm A = \sup \set {\dfrac {\norm {A x}_Y} {\norm x_X}: x \in X, x \ne \mathbf 0_X}$
This supremum is to be taken in $\closedint 0 \infty$ so that $\sup \O = 0$.