Fraktal

Clearly the set of continuous function \( C \left [ a, b \right ] \) is a vector space. Now, we have to prove that the function \( \left | \left | \cdot \right | \right | _{\infty} :  C \left [ a, b \right ] \to \mathbb{R}  \) such that  \( \left | \left | f \right | \right | _{\infty} = \displaystyle \max_{a\leq x\leq b} \left| f(x) \right| \) for every function \( f \in C \left [ a, b \right ] \) is really a norm.

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, || ɑ x || = | 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