Pointwise vs. uniform convergence

In summary, a function is pointwise bounded on a set E if for every x\in E there is a finite-valued function \phi such that |f_n(x)|<\phi(x) for n=1,2,.... A function is uniformly bounded on E if there is a number M such that |f_n(x)|<M for all x\in E, n=1,2,.... The main difference between these two types of boundedness is that in uniform boundedness, the bound is independent of x, while in pointwise boundedness it is dependent on x. This means that for a function to be uniformly bounded, it must have a maximum value that applies to all points in the set E. However
  • #1
ForMyThunder
149
0
A function is pointwise bounded on a set [tex]E[/tex] if for every [tex]x\in E[/tex] there is a finite-valued function [tex]\phi[/tex] such that [tex]|f_n(x)|<\phi(x)[/tex] for [tex]n=1,2,...[/tex].

A function is uniformly bounded on [tex]E[/tex] if there is a number [tex]M[/tex] such that [tex]|f_n(x)|<M[/tex] for all [tex]x\in E, n=1,2,...[/tex].

I understand that in uniform boundedness, the bound is independent of [tex]x[/tex] and in pointwise convergence it is dependent. My question is this: if we take [tex]M=\max\phi(x)[/tex], then since [tex]\phi[/tex] is finite-valued, wouldn't this make every pointwise bounded function a uniformly bounded function? I don't understand.
 
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  • #2
If E is the set (0,1) and [tex]\phi=\frac{1}{x}[/tex]...

I think you can figure out the rest
 
  • #3
Oh, thanks. It said finite-valued which I took to mean bounded.
 
  • #4
I think for domain being a finite set, both notions coincide because maximum is indeed supremem. However, when it is either countable or uncountable domain, it is not necessary to have the equivalence between maximum and supremum. And it can turn out that supremum is unbounded depsite boundedness at each x. Office_Shredder showed a nice example.
 
  • #5
ForMyThunder said:
Oh, thanks. It said finite-valued which I took to mean bounded.

Finite-valued must have merely meant that [itex]|\phi(x)| < \infty[/itex] for all [itex]x[/itex].
 

Related to Pointwise vs. uniform convergence

What is the difference between pointwise and uniform convergence?

Pointwise convergence of a sequence of functions means that for every point in the domain of the function, the sequence of function values converges to the limit function value at that point. Uniform convergence means that the sequence of functions converges to the limit function at every point in the domain simultaneously.

How do you determine if a sequence of functions is pointwise or uniformly convergent?

A sequence of functions is pointwise convergent if, for every point in the domain, the sequence of function values converges to the limit function value at that point. A sequence of functions is uniformly convergent if, for every positive real number ε, there exists a natural number N such that for all n > N and for all x in the domain, the distance between the nth function and the limit function is less than ε.

What is an example of a sequence of functions that is pointwise convergent but not uniformly convergent?

An example of such a sequence is f_n(x) = x^n on the interval [0,1]. This sequence is pointwise convergent to the function f(x) = 0, as for any x in [0,1], the limit of x^n as n approaches infinity is 0. However, this sequence is not uniformly convergent as for any positive real number ε, there is no natural number N such that for all n > N and for all x in [0,1], the distance between f_n(x) and f(x) is less than ε.

Why is uniform convergence considered stronger than pointwise convergence?

Uniform convergence is considered stronger because it ensures that the sequence of functions converges to the limit function at every point in the domain simultaneously, whereas pointwise convergence only guarantees convergence at each individual point. This means that the rate of convergence is the same at every point, making it easier to analyze and work with the limit function.

What are some applications of uniform convergence in mathematics and science?

Uniform convergence is used in many areas of mathematics and science, including analysis, differential equations, and numerical analysis. It is particularly useful in proving the convergence of series and integrals, as well as in approximating functions and solutions to equations. It also plays a key role in the theory of Fourier series and in studying the behavior of power series. In science, uniform convergence is utilized in fields such as physics, chemistry, and engineering to model and understand physical phenomena.

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