Norm

A norm is a function $p:X\to \mathbb{R}$ satisfying

• The triangle inequality: $p(x+y)\le p(x)+p(y)\forall x,y\in X$
• absolute homogeneity: $p(sx)=\Vert s\Vert p(x)\forall x\in X,\forall s\in F$
• point separation: $p(x)=0⟹x=0$

In the above:

• $X$ is a Vector Space
• $F$ is a subfield of the complex numbers $\mathbb{C}$
• $\Vert \cdot \Vert :F\to {\mathbb{R}}_0^{+}$ is the usual norm on $\mathbb{C}$, given by $\Vert a+bi\Vert =a^2+b^2$

Properties

• Norms are non-negative
• Cauchy-Schwarz inequality
• Inner Products define a Canonical norm