Inner Product Norm is Norm

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): $I.1.5$