Fraktal

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

  Let f : ℝ n → ℝ and g : ℝ n → ℝ be scalar-valued differentiable function of several ( n ) variables. Then f and g satisfy the product rule (the Leibniz rule):

  Problem: Let and be and , respectively, such that Determine with proof the matrix .

 Let 𝔽 be any field. A set V with two operation (+:V×V→V called vector addition, and •:𝔽×V→V called scalar multiplication) is a vector space over the field 𝔽 if and only if satisfy all the axioms below.