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.

  It is easy to see that \( \mathrm{arg} \left ( \frac{a}{b} \right ) = \mathrm{arg} \left ( a \right )  - \mathrm{arg} \left ( b \right )  \) for any complex number \( a \) and \( b \). But, this is not the case for the principle value that is \( \mathrm{Arg} \) does not always equal to \( \mathrm{Arg} \left ( a \right ) - \mathrm{Arg} \left ( b \right )  \).

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

 Generate a computable number.

  Problem: Let be a function defined on the real line such that for every . Prove for all , i.e. is constant.   Show Solution Solution: Take any . Next, we obtain for all . So, for , . Then, take the limit as approaches , we get   Hence, the derivative is zero 0 everywhere. Therefore, must be a constant function.

Let I be a closed interval and f : I → ℝ be a differentiable function. Then the derivative f' of f has the intermediate value property : if a , b ∈ I with a < b , then for every y ∈ ℝ such that  f' ( a ) < y < f' ( b ) or f' ( b ) < y < f' ( a ), there exists x ∈ [ a , b ] such that f' ( x ) = y .

Let A be a subset of the set of all complex numbers ℂ and z 0 be an accumulation point of set A . The limit of a function f : A → ℂ as z approaches z 0 is L (denoted ) if for all 𝜀 > 0 there exists 𝛿 > 0 such that if z ∈ A and | z - z 0 | < 𝛿 then | f ( z ) - L | < 𝜀.

Let A be a subset of the set of all real numbers ℝ. A real number c is called the infinum of A ( c = inf A ) if satisfies:

 A sequence of real function ( f n ) is said uniformly convergent on a set A⊆ℝ to a limit function f :A→ℝ if and only if for every 𝜺 > 0 there exists N ∈ ℕ such that for every natural number n ≥ N and for every x ∈ A,