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I would like to appreciate functional analysis in a rigorous setting, to enrich my “engineering” understanding of the matter.

So far I have firmly understood that:

1) for any Banach space a dual space can be defined

2) this brings with itself the notion of weak and weak* convergence.

I think I understand weak convergence.

In vulgar terms, instead of “monitoring” how the series converges in the original space E, it has to be assured that for any element of the dual, the value in R to which the mapping form the dual maps the element in the original space has to converge.

With regards to weak* convergence, I understand that the sequence is defined in the dual space, and the convergence has to be valid when the mapping defined by the sequence is applied to any element of the original space E.

But then, I am not sure I appreciate the consequences or the benefits of this definition.

To start with, I would be the most obliged if somebody could explain ,even with heuristics terms or showing examples, the following qualitative statements I found in literature:

- “<..> to check the weak convergence for a sequence of E, one needs to know what is the space E’. {fair enough}. It may happen that E’ is too big a space. This renders the verification of the weak convergence condition too difficult. Moreover in this case there are too few weakly convergent sequences”

- “The weak topology on X is the weakest topology (the topology with the fewest open sets) such that all elements of X* remain continuous”

Thank you very much

Best Regards

Muzialis

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# Weak and weak* convergence

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