= Norm =

A '''norm''' is a generalized [[Calculus/Distance|distance]].

<<TableOfContents>>

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== Description ==

In Euclidean spaces, the Euclidean distance describes the magnitude of a vector.

The idea of distance is generalized for [[LinearAlgebra/InnerProduct|inner product spaces]] as the '''norm'''. The '''natural norm''' for an inner product space is defined using the inner product: ''||a|| = √⟨a, a⟩''.

This is not the only feasible norm. A norm must satisfy these properties:
 * ''||a|| ≥ 0'' and ''||a|| = 0'' only if ''a'' is the zero vector
 * ''||ca|| = |c| ||a||'' for any scalar c
 * The [[Analysis/CauchySchwarzInequality|triangle inequality]]: ''||a + b|| ≤ ||a|| + ||b||''

Other feasible norms include:
 * The p-norm
   * If ''a = [x y z]'', then ''||a||,,p,, = ^p^√(|x|^p^ + |y|^p^ + |z|^p^)''.
   * Note that Euclidean distance is equivalent to the 2-norm. The absolute value operators are unnecessary for even ''p''s.
 * Max norm
   * If ''a = [x y z]'', then ''||a||,,∞,, = max{|x|, |y|, |z|}''.



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CategoryRicottone