Norm

Topics: Algebra - Vector Space - Inner Product

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Let $(V,⟨,⟩)$ a vector space with inner product on $\mathbb{K}$.

Then, the function $|\cdot |:V\to \mathbb{K}$:

$$\Vert x\Vert :=\sqrt{⟨x,x⟩}$$

…is a norm if, given $x\in V,\lambda \in \mathbb{K}$:

1. $\Vert x\Vert \ge 0$
2. $\Vert x\Vert =0⟺x=0$
3. $\Vert \lambda x\Vert =\overline{\lambda }\Vert x\Vert$
   • Also written $\Vert \lambda x\Vert =|\lambda |\Vert x\Vert$
4. $\Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert$

For example, we can define a norm in the set of real numbers, from where we can also define a metric.