Inner Product Norm is Norm
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Theorem
Let $V$ be an inner product space over a subfield $\Bbb F$ of $\C$.
Let $\norm {\, \cdot \,}$ denote the inner product norm on $V$.
Then $\norm {\, \cdot \,}$ is a norm on $V$.
Proof
Let us verify the norm axioms in turn.
Axiom $(\text N 1)$
\(\ds \) | \(\) | \(\ds x = 0_V\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \) | \(\) | \(\ds \innerprod x x = 0\) | Property $(5)$ of Inner Product | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \) | \(\) | \(\ds \innerprod x x^{1 / 2} = 0\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \) | \(\) | \(\ds \norm x = 0\) | Definition of Inner Product Norm |
$\Box$
Axiom $(\text N 2)$
Part $(2)$ of Properties of Semi-Inner Product.
$\Box$
Axiom $(\text N 3)$
Part $(3)$ of Properties of Semi-Inner Product.
$\Box$
Hence all the properties of a norm have been shown to hold.
$\blacksquare$
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $I.1.5$