Vector Norms
Vector norms
\[
\mathbb{R}_{+} = \left\{ x \in \mathbb{R} \, : \, x\ge 0 \right\} .
\]
Let V be a vector space over either the field of real numbers ℝ or complex numbers ℂ. A norm on V
is a function from a real or complex vector space V to the nonnegative real numbers ℝ+ that satisfies the following conditions:
We mention four of many norms:
- Positivity: ‖v‖ ≥ 0, ‖v‖ = 0 iff v = 0.
- Homogeneity: ‖kv‖ = |k| ‖v‖ for arbitrary scalar k.
- Triangle inequality: ‖v + u‖ ≤ ‖u‖ + ‖v‖.
-
For every x = [x1, x2, … , xn] ∈ 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 ℓp-norm (for p≥1)
\[ \| {\bf x}\|_p = \left( \sum_{k=1}^n x_k^p \right)^{1/p} = \left( x_1^p + x_2^p + \cdots + x_n^p \right)^{1/p} . \]
Theorem 3:
The following inequalities hold for all x ∈ ℂn or x ∈ ℝn:
====================================
- \( \displaystyle \| {\bf x} \|_{\infty} \le \| {\bf x} \|_{1} \le n\,\| {\bf x} \|_{\infty} , \)
- \( \displaystyle \| {\bf x} \|_{\infty} \le \| {\bf x} \|_{2} \le \sqrt{n}\,\| {\bf x} \|_{\infty} , \)
- \( \displaystyle \| {\bf x} \|_{2} \le \| {\bf x} \|_{1} \le \sqrt{n}\,\| {\bf x} \|_{2} .\)
including a special class of vecors, known as Euclidean space. Their properties were collected by the ancient Greek mathematician Euclid in his Elements.
In given any (real or complex) vector space V, two norms
‖ ‖a and ‖ ‖b are equivalent if and only if (iff) there exists some positive constants c1 and c2 such that
\[
\| {\bf x}\|_a \le c_1 \| {\bf x}\|_b \qquad \mbox{and} \qquad \| {\bf x}\|_b \le c_2 \| {\bf x}\|_a \qquad \mbox{for all } \quad {\bf x} \in V.
\]
Theorem 4:
If V is any real or complex vector space of finite dimension, then any two norms on V are equivalent.
With dot product, we can assign a length of a vector, which is also called the Euclidean norm or 2-norm:
\[
\| {\bf x} \|_2 = \sqrt{ {\bf x}\cdot {\bf x}} = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2} .
\]
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.
On an n-dimensional complex space \( \mathbb{C}^n ,\) the most common norm is
\[
\| {\bf z} \| = \sqrt{ {\bf z}\cdot {\bf z}} = \sqrt{\overline{z_1} \,z_1 + \overline{z_2}\,z_2 + \cdots + \overline{z_n}\,z_n} = \sqrt{|z_1|^2 + |z_2 |^2 + \cdots + |z_n |^2} .
\]
