Norms

Why

We generalize the concept of norm from real vectors to abstract vector spaces.

Definition

A norm is a real-valued functional that is (a) non-negative, (b) definite, (c) absolutely homogeneous, (d) and satisifies a triangle inequality. The triangle inequality property requires that the norm applied to the sum of any two vectors is less than the sum of the norms on those vectors.

A normed space (or norm space) is an ordered pair: a vector space whose field is the real or complex numbers and a norm on the space. We require the vector space to be over the field of real or complex numbers because of absolute homogeneity: the absolute value of a scalar must be defined.

Notation

Let $(X, \F )$ be a vector space where $\F $ is the field of real numbers or the field of complex numbers. Let $f: X \to \R $. The functional $f$ is a norm if

1. $f(v) \geq 0$ for all $x \in V$
2. $f(v) = 0$ if and only if $x = 0 \in X$.
3. $f(\alpha x) = \abs{\alpha }f(x)$ for all $\alpha \in \F $, $x \in X$
4. $f(x + y) \leq f(x) + f(y)$ for all $x, y \in X$.

In this case, for $x \in X$, we denote $f(x)$ by $\norm{x}$, read aloud “norm x”. The notation follows the notation for the absolute value function is a norm on the vector space of real numbers. In some cases, we go further, and for a norm indexed by some parameter $\alpha $ or set $A$ we write $\norm{x}_\alpha $ or $\norm{x}_A$.

When the field is assumed or clear from context, it is succinct to say let $(V, \norm{\cdot })$ be a normed space.

Examples

The absolute value function is a norm on the vector space of real numbers. In addition, the (Euclidean norm) is a norm on the vector space $\R ^n$.

Other terminology

The descriptive but slight more verbose norm vector space and normed vector space are also in usage.