Norms

In order to define how close two vectors are, and in order to define the convergence of sequences of vectors, mathematicians use a special device, called metric (which is actually a distance). However, since we need to incorporate vector additions, a metric in vector spaces is generated by a norm.

Let V be a vector space over a field 𝔽, where 𝔽 is either the field ℝ of reals, or the field ℚ of rational numbers, or the field ℂ of complex numbers. A norm on V is a function ∥ ∥ : V → ℝ₊ = { r∈ℝ : r ≥ 0 }, assigning a nonnegative real number ∥u∥ to any vector u ∈ V, and satisfying the following conditions for all x, y, z ∈ V:

1. Positivity:    ∥x∥ ≥ 0 and ∥x∥ = 0 if and only if x = 0.
2. Homogeneity:     ∥kx∥ = |k|·∥x∥.
3. Triangle inequality:    ∥x+y∥ ≤ ∥x∥ + ∥y∥.

A vector space V together with a norm ∥·∥ is called a normed vector space.

Out of many possible norms, we mention four the most important norms:

• For every x = [x₁, x₂, … , x_n] ∈ V, we have the 1-norm:
  $$\| {\bf x}\|_1 = \sum_{k=1}^n | x_k | = |x_1 | + |x_2 | + \cdots + |x_n |.$$
  It is also called the Taxicab norm or Manhattan norm.
• The Euclidean norm or ℓ²-norm is
  $$\| {\bf x}\|_2 = \left( \sum_{k=1}^n x_k^2 \right)^{1/2} = \left( x_1^2 + x_2^2 + \cdots + x_n^2 \right)^{1/2} .$$
• The Chebyshev norm or sup-norm ‖v‖_∞, is defined such that
  $$\| {\bf x}\|_{\infty} = \max_{1 \le k \le n} \left\{ | x_k | \right\} .$$
• The ℓ^p-norm (for p≥1)
  $$\| {\bf x}\|_p = \left( \sum_{k=1}^n x_k^p \right)^{1/p} = \left( x_1^p + x_2^p + \cdots + x_n^p \right)^{1/p} .$$

With dot product, we can assign a length of a vector, which is also called the Euclidean norm or 2-norm:

Since this definition involves a square root, we always use a positive branch of the analytic function $\sqrt{x} .$ Namely, we choose the branch that assigns $\sqrt{x} > 0$ for z > 0.

Dot product is a particular case of more general bilinear form, known as inner product. An inner product space is a vector space with an additional structure called an inner product. So every inner product space inherits the Euclidean norm and becomes a metric space. In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero.

The definition of norm in ℂⁿ needs an accormodation of complex conjugate numbers that are denoted either by overline (mathematics) or asterisk (physics and engineering):

where j is the imaginary unit on the complex plane ℂ, so j² = −1. Then the Euclidean norm on an n-dimensional complex space is defined by