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Beginning functional analysis I thought I would learn about generalized properties of functions and operators but yet I am flipping through pages after pages of texts on cauchy sequence, convergence, etc.

What is the purpose of learning about sequences in functional analysis? Seems rather pointless to me as I have already learned about sequences in elementary calculus

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    Local enumerable basis is the answer. If you want to study more general spaces, look at topological vector spaces. – user40276 Jul 11 '15 at 06:20
  • I would say that one good definition of Calculus is that it is the study of limits and limiting processes. As such, sequences play a pivotal role in everything that you'll do in Analysis. – Disintegrating By Parts Jul 11 '15 at 06:25

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Sequences are a fundamental way of probing topological spaces in general, e.g. in investigating whether a given set is compact, a question which is of interest in functional analysis (see the Banach-Alaoglu theorem, and also the thread Why is compactness so important?). However, you'll also find many presentations of functional analysis that instead use a generalization of the notion of sequence, called a "net" (Wikipedia link).

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Zev Chonoles
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