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after studying summable series, I know that there exists the concept of summable families. There is any book where I can find the proof of the basic properties of summable families?

Thanks.

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Summable families in what sense? Something like $$\sum_{j \in J} a_j$$where $J$ is uncountable? Why would there be a book on that? At most there would be a few remarks within a textbook where the countable case is considered more fully. One such is Exercise 6 S (p. 214) in

Kelley, John L., General topology, The University Series in Higher Mathematics. New York: D. van Nostrand Co., Inc. 298 p., illustr. (1955). ZBL0066.16604.

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