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Can you recommend me books in which i can learn functional analysis and understand material like weak convergence (with its topology) ? what are also the first books written on the subject. Is it Banach's book "operator theory" ?

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I haven't read either of these, but I know that both Serge Lang's Real and Functional Analysis and Rudin's Functional Analysis cover weak convergence.

I don't know anything about the history of the subject, but Rudin gives a brief discussion on the importance of Banach's work in the appendix of his text.

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