11.2 Normed vector space

Now we equip a vector space $V$ with a notion of "size".

• A norm is a function ( $:((\cdot ), V\to \mathbb{R})$ ) such that the following properties hold ( $\forall x, y, x\in V\land y\in V$ and $\forall \alpha , \alpha \in $ ):
  • $(x)\ge 0$ with equality iff $x=0$
  • $(\alpha x)=\left|\alpha \right|\cdot (x)$
  • $(x+y)\le (x)+(y)$ , (the triangle inequality ).
  In simple terms, the norm measures the size of avector. Adding the norm operation to a vector space yields a normed vector space . Important example include:
  • $V=\mathbb{R}^N$ , $≔((\left(\begin{array}{c}{x}_0\\ \dots \\ x_{N-1}\end{array}\right)), \sqrt{\sum_{i=0}^{N-1} x_i^2}, \sqrt{x^Tx})$
  • $V=\mathbb{C}^N$ , $≔((\left(\begin{array}{c}{x}_0\\ \dots \\ x_{N-1}\end{array}\right)), \sqrt{\sum_{i=0}^{N-1} \left|x_i\right|^2}, \sqrt{(x)x})$
  • $V=l_p$ , $≔((\{x(n)\}), \sum_{n=()} ^{})$∞ ∞ x n p 1 p
  • $V={ℒ}_p$ , $≔((f(t)), \int_{()} \,d t^{})$∞ ∞ f t p 1 p