Norm on Bounded Linear Transformation is Submultiplicative

Theorem

Let $\struct {X, \norm \cdot_X}$, $\struct {Y, \norm \cdot_Y}$ and $\struct {Z, \norm \cdot_Z}$ be normed vector spaces.

Let $A : X \to Y$ and $B : Y \to Z$ be bounded linear transformations.

Let $\norm \cdot _{\map B {X,Y} }$ be the norm for bounded linear transformations $X \to Y$.

Let $\norm \cdot _{\map B {Y,Z} }$ be the norm for bounded linear transformations $Y \to Z$.

Let $\norm \cdot _{\map B {X,Z} }$ be the norm for bounded linear transformations $X \to Z$.

Then, we have that:

$B \circ A$ is a bounded linear transformation

with:

$\norm {B \circ A} _{\map B {X,Z} } \le \norm B _{\map B {Y,Z} } \norm A _{\map B {X,Y} }$

That is:

$\norm \cdot$ is submultiplicative.

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