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Section 2.5 Norms

In a vector space it turns out to be very useful being able to associate a length to vectors. This is done through the introduction of a norm as follows.
Definition 2.5.1. Norm.
Let \(V\) be a vector space over a field \(\bK\text{.}\) A norm on \(V\) is a map \(\|\cdot\|:V\times V\to[0,\infty)\) such that:
  1. \(\|v\|=0\) if and only if \(v=0\text{;}\)
  2. \(\|\lambda v\|=|\lambda|\cdot\|v\|\) for all \(\lambda\in\bK\) and \(v\in V\text{;}\)
  3. \(\|u-v\|\leq\|u\|+\|v\|\) for all \(u,v\in V\text{.}\)
A norm \(\|\cdot\|\) makes \(V\) a metric space with \(d(u,v)=\|u-v\|\) and so a (Hausdorff) topological space. In fact, with such topology \(V\) is a topological vector space (see Exercise 2.7.4).
Definition 2.5.2. Equivalent norms.
Two norms on \(V\) are said equivalent if they give rise to homeomorphic topological structures on \(V\text{.}\)
On a vector space there can be many norms. In finite dimension, the following norms are often used:
  1. \(\|v\|_2=\sqrt{(v^1)^2+\dots+(v^n)^2}\) (Euclidean norm);
  2. \(\|v\|_1=|v^1|+\dots+|v^n|\) (Taxicab norm);
  3. \(\|v\|_\infty=\max(|v^1|,\dots,|v^n|)\) (Chebyshev norm).
  4. \(\|v\|_p=\left(|v^1|^p+\dots+|v^n|^p\right)^{1/p}\) (\(\ell_p\) norm), \(p\geq1\text{;}\)

The following code shows the unit circles for some of the norms above:

In general, different norms give rise to inequivalent topologies. In finite dimension, though, the situation is very simple: In fact, more can be proved: