Uniform Convergence - Questions Answered

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Discussion Overview

The discussion revolves around questions related to uniform convergence of sequences of functions, particularly in the context of continuity and the relationship between pointwise and uniform convergence in various metric spaces. Participants explore implications for piecewise continuous functions and the conditions under which pointwise convergence may be equivalent to uniform convergence.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether the uniform limit of piecewise continuous functions is also piecewise continuous, given that the functions may be discontinuous at different points.
  • Another participant notes that in certain L^p spaces, pointwise convergence can be equivalent to uniform convergence under specific conditions, such as the presence of an integrable function that bounds the sequence.
  • A later reply references the Dominated Convergence Theorem as relevant to the discussion of convergence properties.
  • One participant argues that uniform convergence cannot always be equated with convergence in the norm, particularly in the context of sequences of sequences, and provides a counterargument to support this claim.
  • Another participant defines norm convergence in a Hilbert space, indicating a specific mathematical framework for discussing convergence.

Areas of Agreement / Disagreement

Participants express differing views on the implications of uniform convergence for piecewise continuous functions and the equivalence of pointwise and uniform convergence in various contexts. The discussion remains unresolved regarding the general conditions under which these properties hold.

Contextual Notes

Limitations include the dependence on definitions of continuity and convergence, as well as the specific properties of the metric spaces being considered. The discussion does not resolve the mathematical steps involved in the arguments presented.

T-O7
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Hello,

I have two questions to ask regarding uniform convergence for sequences of functions.

So I know that if a sequence of continuous functions f_n : [a,b] -> R converge uniformly to function f, then f is continuous.

Is this true if "continous" is replaced with "piecewise continuous"? (I am not assuming that the sequence functions are discontinuous at the same points)
i.e. if f_n are each discontinuous at finitely many points, is the uniform limit function f discontinuous at finitely many points as well?

Also, does anyone know for what kinds of metric spaces (if any) is "pointwise convergence" for sequences of functions equivalent to "uniform convergence"?

Thanx.
 
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T-O7 said:
Also, does anyone know for what kinds of metric spaces (if any) is "pointwise convergence" for sequences of functions equivalent to "uniform convergence"?

Thanx.
Using the L^p spaces have this property in a sense under certain restrictions (such has having an integrable function that is always greater than the sequence).
 
e.g. Dominated Convergence Theorem
 
I'd like to add that something, but it has to do with functional analysis. If you were to consider "uniform convergence" the same thing as convergence in the norm, then there are cases where this cannot be true. In some cases, pointwise convergence cannot be the same as convergence in the norm (ie. there is no norm, that would have those qualities).

One example would be the set of all sequences (they don't need to be bounded) with either real or complex numbers. There is no norm that would have the following property: a sequence of sequences converges converges pointwise if and only if the sequence converges in the norm. The reason that is not possible can be proven pretty simply by a counterargument.

(Outline of Proof: Assume such a norm exists. For any bounded sequence of sequences, you can then choose a subsequence by induction in each variable--ie. the i-th place of the sequence--so that they converge because each component is bounded sequence of real or complex numbers. The subsequence chosen would then converge pointwise. But that means that it also converges in the norm. What that implies is that every bounded sequence has a convergent subsequence in this norm. That implies that the dimension of the set off all sequences is finite. This is obviously not true and therefore there cannot exist a norm with the mentioned properties.)

So basically it depends on what you mean by "convergence in the norm". Under certain restrictions it can be realized, but in many cases a general convergence from pointwise convergence cannot be realized (which is where topological vector spaces take over for normed vector spaces).
 
Norm (or strong) convergence of, say, a sequence of vectors {xn} in a Hilbert space may be defined thus:

[tex]x_{n}\rightarrow x\mbox{ if, and only if }\|x_n - x\|\rightarrow 0\mbox{ as } n\rightarrow\infty[/tex]
 

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