Bounded Linear Norm
Why
We can give the set of bounded linear functions between two norm spaces a norm.
Definition
The norm of a bounded linear function is the smallest real number by which we can bound the result on a vector times the norm of that vector.
Notation
Let $((V_1, F_1), \norm{\cdot }_1)$ and $((V_2,
F_2), \norm{\cdot }_2)$ be two norm spaces.
Let $f: V_1 \to V_2$ be linear and bounded.
The norm of $f$ is the smallest $C$ so that
\[
\norm{f(v)}_2 \leq C\norm{v}_1.
\]
Equivalent formulation
Let $((V_1, F_1), \norm{\cdot }_1)$ and $((V_2,
F_2), \norm{\cdot }_2)$ be two norm spaces.
Let $f: V_1 \to V_2$ bounded and linear.
The norm of $f$ is
\[ \sup_{\norm{x}_1 = 1} \norm{f(x)}_2. \]