The Connections Project

Real Analysis

The complex plane is a generalization of the real plane in the most obvious way. The latter is embedded in the former. Many functions in real analysis, such as the exponential function and the sine function, hide profoundly deep properties until they are extended in complex analysis.

Various structures studied in real analysis have inherent algebraic structure as well. Thus, the former profits from an understanding of the latter, while the latter can draw many interesting examples from the former.

Some basic topology is absolutely essential to understand real analysis, to the extent that most textbooks on analysis actually include a whole chapter on topology. Concepts like "open interval", "limit point", and so on, of which we learn naive versions in calculus, need the machinery of abstract topology to generalize. Some topological notions like "simply connected" (naively "connected with no holes") look deceptively simple and yet without some rather advanced topology, it's almost impossible to rigorously define them.

Conversely, the real line and Euclidean space provide so many natural examples in topology that a topology textbook without lots of analytical examples would be like a barren wasteland.

In one direction, set theory is needed to actually construct the real numbers: without set theory, analysis is just a kind of speculation: "*IF* some set of objects exists with all these wonderful properties, *THEN*..." Some interesting set theory is needed to actually prove that "the reals exist" (i.e., that some ordered field with sups and infs exists).

Going into more advanced territory, set theory provides Zorn's Lemma (which is equivalent to the Axiom of Choice), which analysts invoke constantly.

Going the other way, the reals provide a very natural and intuitive playground for one of the deepest issues in set theory, the Continuum Hypothesis.

The machinery of real analysis can often be used to prove certain facts in number theory. For example, if you want to demonstrate that a certain number theoretic function is increasing, if it has an obvious extension to the whole real line, you can try showing the extension has a positive derivative.

In turn, the real analyst can make herself more cultured by studying certain generalizations of the reals which generally fall in the realm of number theory: for example, the completions of the p-adic numbers.

One of the biggest, least expected connections between these fields comes from the DPRM-Theorem, also known as The Solution to Hilbert's 10th Problem. This theorem says that the recursively enumerable sets are precisely the diophantine sets. This can be used to prove some otherwise intractable statements in real analysis which don't appear to be related to computability theory at all.

In turn, real analysis contributes back to computability theory by providing concepts like "computable real number", "computable real function", and so on, which is collectively the subject of a subfield sometimes called "computable real analysis".

The connection between graph theory and real analysis arises when the planarity of graphs is considered. A planar graph is one which can be embedded into 2-dimensional Euclidean space such that no two edges cross. Considered as abstract vertex- and point-sets, it makes no sense to talk about edges intersecting each other.

Analysis is chock-full of vector spaces, often infinite-dimensional ones such as function spaces and functional spaces.