What Does Uniform Convergence Mean With it's application

Uniform Convergence :

       Uniform Convergence is a part of real analysis which is discussed in detail below.

Definition :

            Let (X,d) be a metric space and f be a function from X to R . Also for each n ∈ ℕ let fₙ : X ---> R . Then , the sequence of functions <fₙ> converges pointwise to the function f , if for each x ∈ X , the sequence of real numbers < fₙ(x) > converges to the real number f(x) . 

                 Therefore <fₙ(x) > converges pointwise to f if  lim   fₙ(x) = f(x)    ∀ x ∈ X .

                             n-->∞

For Example : 

          Let < fₙ > be the sequence defined by fₙ  : R --> R such that fₙ(x) = x / n  ∀ x∈R , n∈N  . Show that the sequence converges pointwise to zero function .

Solution :

         Here , we have to show that the given sequence converges pointwise to the zero function i.e f(x) = 0 , x∈ R , then we must show that given ε > 0 ,we can find m∈ ℕ  such that 

      ∀ n ≥ m ⇒ | x/n - 0 | = |x| / n ....(1)

Let us choose m> |x|/ε 

Then equation (1) gives ∀ n≥ m

      ⇒ |x/n - 0| = |x|/n < ε

Here , the given sequence converges pointwise to zero function .

Uniform Convergence Of A Series Of Functions :

Definition :

                               ∞

            The series Σ uₙ(x) is said to converge 

                              n=1

uniformly on X if the sequence <fₙ(x)> , where fₙ(x) = u₁(x)+u₂(x)+.......+uₙ(x) converges uniformly on f .

                               ∞

                   where Σ uₙ(x) is called 

                               n=1

sum function of series .