Definition of Normed Space

A normed space or a normed vector space is a vector space over real or complex numbers on which a norm is defined. A norm is a real-valued function ||•|| defined on a vector space V with scalars in a field 𝔽 (the real numbers or the complex numbers) such that satisfies following properties:

1. ||x|| = 0 only if x = 0.

2. For every x ∈ V and ɑ ∈ 𝔽, ||ɑx|| = |ɑ| ||x||.

3. For every x,y ∈ V, ||x + y|| ≤ ||x|| + ||y|| (triangle inequality).

Maybe someone would be thinking the property "||x|| ≥ 0 for all x ∈ V (nonnegative)" must be included in the definition. This property actually can be proved by using the three other properties.

Example

 On the set of continous function $C[a,b]$, we can define a norm ${\left||\cdot |\right|}_{\mathrm{\infty }}$ which given by

 ${\left|\left|f\right|\right|}_{\mathrm{\infty }}={\displaystyle \underset{a\le x\le b}{max}|f(x)|}$ for every function $f\in C[a,b]$