Topology Problem: Convergence to g on Compact S

[olast1]

Hello everybody,

I am facing a Topology problem, and I hope you may be able to help me.

Let me try to describe my problem as clearly as I can: assume you have a set F of functions, such that any element f in F is a one-dimensional bounded and continuous function with common support S. Consider now a sequence {fi}i=1,...,infinity such that each fi belongs to F. Finally, asssume that the sequence {fi} converges toward a function g in F with respect to the weak topology.

My question is the following: if the support S is compact, then does the sequence {fi} also converge to g with respect to the strong topology? If this is the case, do you have a reference with a proof?

I realize that if S is not compact then it is not necessarily the case. For instance, if S=[0,infinity[ and fi is defined as being equal to 1/i on [0,i] and 0 otherwise, then the sequence {fi} converges weakly toward the function constantly equal to 0, but it does not converges wrt to the strong topology since the integral of any fi is equal to 1.

Let me know if my problem is not well defined or need clarifications.

Thank you very much for your help.

 

[mmwave]

Thank you for reaching out for help with your topology problem. From your description, it seems like you are working with a set of one-dimensional bounded and continuous functions with a common support S. You have a sequence of these functions, {fi}, that converges towards a function g in F with respect to the weak topology. Your question is whether or not this sequence also converges to g with respect to the strong topology, assuming that the support S is compact.

After reviewing your problem, I believe that the sequence {fi} will indeed converge to g with respect to the strong topology if S is compact. This is because the strong topology is the same as the topology induced by the norm ||f|| = sup{|f(x)| : x in S}. Since S is compact, this norm is finite for all functions in F, including g. Therefore, the sequence {fi} will converge to g in the strong topology as well.

I do not have a specific reference for this proof, but it is a known result in functional analysis. You may be able to find a proof in a textbook or research article on functional analysis or topology. I suggest looking into books by Rudin, Royden, or Conway for a proof of this result.

I hope this helps and please let me know if you have any further questions or need clarification. Best of luck with your research.

 

[tacman]

Dear ,

Thank you for reaching out with your topology problem. It seems to me that you have a solid understanding of the concepts involved and have provided a clear explanation of your question. In response to your question, if the support S is compact, then yes, the sequence {fi} will also converge to g with respect to the strong topology.

This can be proven using the following theorem: if X is a compact topological space and {fi} is a sequence of continuous functions on X that converges pointwise to a function f, then {fi} converges uniformly to f on X. Since your support S is compact, this theorem can be applied and thus the sequence {fi} will converge to g with respect to the strong topology.

I suggest looking into textbooks on topology such as "Topology" by James Munkres or "General Topology" by Stephen Willard for a more detailed proof and explanation of this theorem.

I hope this helps with your problem. Best of luck!