Properties of Semi-Inner Product

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

Let $\left \langle{\cdot, \cdot}\right \rangle$ be a semi-inner product on $V$.

Denote, for $x \in V$, $\left\Vert{x}\right\Vert := \left\langle{x, x}\right\rangle^{1 / 2}$.

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


 * $(1): \quad \left\Vert{x + y}\right\Vert \le \left\Vert{x}\right\Vert + \left\Vert{y}\right\Vert$
 * $(2): \quad \left\Vert{a x}\right\Vert = \left|{a}\right| \left\Vert{x}\right\Vert$

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.