I have essentially taught myself everything I know about analysis to suit my needs (and I'm probably worse off for it). I really wish there were a book like "Analysis from a computational perspective," which I suppose is just numerical analysis but I have yet to find any books that suit me in that topic either. That being said, something like Christianini's "Introduction to Support Vector Machines" has doubled as a synthesizing text on basic functional analysis for me. My recommendation, if you're comfortable with proofs that you would see in abstract algebra, is to jump right into baby Rudin or any other undergrad-level analysis text. I view them all pretty much the same. Likewise I essentially learned all the probability theory and combinatorics I know from people and scraps, so I can't recommend a synthesizing text.…
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