Norm of Hermitian Operator/Corollary
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Corollary to Norm of Hermitian Operator
Let $\mathbb F \in \set {\R, \C}$.
Let $\HH$ be a Hilbert space over $\mathbb F$.
Let $A : \HH \to \HH$ be a bounded Hermitian operator.
Let $\innerprod \cdot \cdot_\HH$ denote the inner product on $\HH$.
Suppose that:
- $\forall h \in \HH: \innerprod {A h} h_\HH = 0$
Then $A$ is the zero operator $\mathbf 0$.
Proof
Let $\norm \cdot_\HH$ denote the inner product norm on $\HH$.
Let $\norm A$ denote the norm of $A$.
From Norm of Hermitian Operator:
- $\norm A = \sup \set {\size {\innerprod {A h} h_\HH}: h \in \HH, \norm h_\HH = 1}$
By definition of inner product norm:
- $\forall h \in \HH: \innerprod {A h} h_\HH = 0$
Hence, in particular:
- $\innerprod {A h} h_\HH = 0$
for all $h \in \HH$ such that $\norm h_\HH = 1$.
So:
- $\set {\size {\innerprod {A h} h_\HH}: h \in \HH, \norm h_\HH = 1} = \set 0$
giving:
- $\norm A = \sup \set 0$
Hence from the definition of supremum:
- $\norm A = 0$
So from Norm on Bounded Linear Transformation equals Zero iff Zero Operator:
- $A$ is the zero operator.
$\blacksquare$
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {II}.2.14$