9: Back to the Real Numbers
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- 9.1: Trigonometric Series
- As we have seen, when they converge, power series are very well behaved and Fourier (trigonometric) series are not necessarily. The fact that trigonometric series were so interesting made them a lightning rod for mathematical study in the late nineteenth century.
- 9.2: Infinite Sets
- All of our efforts to build an uncountable set from a countable one have come to nothing. In fact many sets that at first “feel” like they should be uncountable are in fact countable. This makes the uncountability of R all the more remarkable. However if we start with an uncountable set it is relatively easy to build others from it.
- 9.3: Cantor’s Theorem and Its Consequences
- Once Cantor showed that there were two types of infinity (countable and uncountable), the following question was natural, “Do all uncountable sets have the same cardinality?”
Thumbnail: Georg Cantor, German mathematician and philosopher of mixed Jewish-Danish-Russian heritage, the creator of set theory. (public domain).