Norm

A norm is a generalized distance.

Description

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

The idea of distance is generalized for 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

• ||ca|| = |c| ||a|| for any scalar c

• ||a + b|| ≤ ||a|| + ||b||

The third property is called the triangle inequality, and it follows from the Cauchy-Schwartz inequality (⟨u, v⟩² ≤ ⟨u, u⟩⟨v, v⟩) which itself is a property of inner products.

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 ps.

• Max norm

  • If a = [x y z], then ||a||_∞ = max{|x|, |y|, |z|}.

CategoryRicottone