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exchanging integrals, sums, or limits, or integrating by parts) that one wishes to perform.Once one obtains these estimates, one can then often take some sort of limiting argument to recover the general case. If one needs to flip an upper bound to a lower bound or vice versa, look for a way to take reflections or complements.
Similarly: In order to pass back to the general case from these special cases, one will have to somehow decompose the general object into a combination of special ones, or approximate general objects by special ones (or as a limit of a sequence of special objects).Sometimes one needs a lower bound for some quantity, but only has techniques that give upper bounds.In some cases, though, one can “reflect” an upper bound into a lower bound (or vice versa) by replacing a set contained in some space with its complement , or a function with its negation (or perhaps subtracting from some dominating function to obtain ).Uncountable unions are not well-behaved in measure theory; for instance, an uncountable union of null sets need not be a null set (or even a measurable set).(On the other hand, the uncountable union of sets remains open; this can often be important to know.) However, in many cases one can replace an uncountable union by a countable one.But I had not thought to actually try to make these tricks explicit, so I am going to try to compile here a list of some of these techniques here.But this list is going to be far from exhaustive; perhaps if other recent students of real analysis would like to share their own methods, then I encourage you to do so in the comments (even – or especially – if the techniques are somewhat vague and general in nature).Similarly, the Cauchy-Schwarz inequality can flip a upper bound on to a lower bound on , provided that one has a lower bound on .Holder’s inequality can also be used in a similar fashion. Uncountable unions can sometimes be replaced by countable or finite unions.&eisbn=978-1-4704-2124-3&pisbn=978-0-8218-2050-6&epc=STML/4. E&ppc=STML/4&title=Problems in Mathematical Analysis I: Real Numbers, Sequences and Series&author=W. In class, I mentioned that when solving the type of homework problems encountered in a graduate real analysis course, there are really only about a dozen or so basic tricks and techniques that are used over and over again.