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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.

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# Topology Question

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