Overview
A vector norm is an abstraction of the concept of the length of a vector in Euclidian space.
{% length \; of \; \vec{v} = \sqrt{v_1 ^2 + ... + v_n^2} %}
Definition
A norm is a function (denoted {% ||x|| %}) that takes a vector, and returns a non-negative real number.
- {% ||x|| \geq 0 %}
- {% ||x|| = 0 %} iff {% x=0 %}
- {% ||\lambda x || = |\lambda| || x || %}
- {% || x + y || \leq ||x|| + ||y|| %}
Normed Linear Space
A normed linear space is a vector space equipped with a vector norm. Having a vector norm induces a topology on the space by using the norm as a metric. That is, the metric is defined as
{% d(x,y) = || x-y || %}
For a linear transformation {% T %} between two normed linear spaces, the following are equivalent
- {% T %} is continuous at a point
- {% T %} is continuous at all points
- {% T %} is bounded
Example Norms
{% ||x||_1 = \sum | x_i | %}
{% ||x||_2 = \sqrt{\sum | x_i |^2} %}
{% ||x||_p = (\sum | x_i |^p)^{\frac{1}{p}} %}
{% ||x||_{\infty} = max_i | x_i | %}