Properties of Semi-Inner Product

Theorem
Let $V$ be a vector space over $\Bbb F \in \set {\R, \C}$.

Let $\innerprod \cdot \cdot$ be a semi-inner product on $V$.

Denote, for $x \in V$, $\norm x := \innerprod x x^{1 / 2}$.

Then, $\forall x, y \in V, a \in \Bbb F$:


 * $(1): \quad \norm {x + y} \le \norm x + \norm y$
 * $(2): \quad \norm {a x} = \size a \norm x$

Proof of $(1)$
For $x, y \in V$, compute:

Taking square roots on either side gives the result.

Proof of $(2)$
For $x \in V$, $a \in \Bbb F$, compute:

Taking square roots on either side gives the result.