Equivalence of Definitions of Norm of Linear Transformation/Definition 4 Greater or Equal Definition 2

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Theorem

Let $H, K$ be Hilbert spaces.

Let $A: H \to K$ be a bounded linear transformation.

Let:

$\lambda_2 = \sup \set {\dfrac {\norm {A h}_K} {\norm h_H}: h \in H, h \ne 0_H}$

and

$\lambda_4 = \inf \set {c > 0: \forall h \in H: \norm {A h}_K \le c \norm h_H}$


Then:

$\lambda_4 \ge \lambda_2$

Proof

From Operator Norm is Finite:

$\lambda_4 < \infty$


From Submultiplicity of Operator Norm:

$\forall h \in H : \norm{A h}_K \le \lambda_4 \norm h_H$

Hence:

$\forall h \in H, h \ne 0_H : \dfrac {\norm{A h}_K} {\norm h_H} \le \lambda_4$

From Continuum Property:

$\lambda_2 = \sup \set {\dfrac {\norm {A h}_K} {\norm h_H}: h \in H, h \ne 0_H}$ exists

By definition of the supremum:

$\lambda_2 \le \lambda_4$

$\blacksquare$