Norm

Given a vector space V, a norm on V is a function $||\cdot ||$: V$\to \mathbb{R}$ such that:

1. Non-degenerate $||x||\ge 0\mathrm{\forall }x\in V$ and $||x||=0⟺x=0$
2. Homogeneity $||\alpha x||=|\alpha |||x||$ $\mathrm{\forall }\alpha \in \mathbb{R},x\in V$
3. Triangle Inequality $||x+y||\le ||x||+||y||,\mathrm{\forall }x,y\in V$

Thm: Inner Product space to Normed Space

If (V,$⟨\cdot ,\cdot ⟩$) is an inner-product space, then defining $||\cdot ||:V\to \mathbb{R}$ as $||v||=\sqrt{⟨v,v⟩}$ gives a norm making (V,$||\cdot ||$) into a normed vector space

*Not all normed vector spaces have an inner product space

Def: Normal

$||v||=1$