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I have sometimes overheard people using the terms hard analysis and soft analysis.I am not a particularly well-read person in mathematics but I have wondered what that is all about.I hope there exists an explanation for someone with single-variable calculus background.

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Terry Tao has an excellent blog post on this topic. –  Bob Pego May 5 '12 at 20:16
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This is more for those who already know the difference. I saw the following joke on the internet sometime in the late 1990s and used it in a MAA talk I gave on non-constructive proofs in 2001. Question: How many analysts does it take to screw in a light bulb? Answer: Three. One to prove existence, one to prove uniqueness, and one to devise a non-constructive way to do it. –  Dave L. Renfro May 7 '12 at 16:04

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Roughly speaking, if you use "functional analysis" methods it is called soft, whereas if you use "estimates" it is called hard.

For example, Weierstrass constructed an example of a function $f \colon [0,1] \to \mathbb R$ that is continuous but nowhere differentiable. His proof involved computing inequalities to show his function was not differentiable. HARD

Nowadays a modern mathematician may consider the Banach space $C[0,1]$ and cite the Baire category theorem to show that there is a function $f \colon [0,1] \to \mathbb R$ which is continuous but nowhere differentiable. SOFT

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Are there any advantages or disadvantages associated with either of the approaches you know of ? Care to share it would be much appreciated. –  Hardy May 5 '12 at 13:40
    
@Hardy If you want to something happens and you give a constructive proof, it is usually more useful in applications. For instance, Banach's fixed point theorem does this by constructing the fixed point. Sometimes it is easier to show existence without giving an example as in the case above. However, it is not particularly useful in constructing functions which are nowhere differentiable to study. This comment was really late now that I'm looking at it. –  Eoin Jun 12 at 5:23

Let me first digress a little to illuminate the big picture. In the mathematical community it seems Tao's blog entry is the most prominent source for this distinction, although it is also a interesting topic for "foundations", to be more precise, for proof theory (Proof theory is a field of mathematical logic where one studies mathematical proofs as (precisely defined) objects, where, e.g., one utilizes tree-like representations for proofs, and considers ways to convert one deduction system to another).

Apart from being part of mathematical logic, proof theory is also significant for mathematics, for it can be applied to actual mathematical proofs. "[T]he logical analysis of proofs using techniques from proof theory" is called unwinding proofs or proof mining ([Koh07], p. 144). Proof mining can be applied to various fields of mathematics, e.g. numerical analysis, ergodic theory and number theory; and the main idea is that by examining the proof of some result we can obtain more effective data related to the original result, such as constructions (of an example, such as Weierstrass' function in Prof. Edgar's answer), rates of convergence, various levels of dependence of solutions to initial parameters etc.. The logical significance of proof mining is mostly due to the endeavor of actuating the so-called Hilbert's program: "How is it that abstract methods ('ideal elements') can be used to prove 'real' statements e.g. about the natural numbers and is this use necessary in principle?" ([Koh07], p. 143). This program is found to be impossible in most cases, but there are partial realizations, and here proof mining comes in play.

Returning to the difference between hard and soft analysis, note first that "soft" and "hard" does not denote two clean-cut ways of doing analysis, instead they denote the two ends of the spectrum of doing analysis as regions, for there are notations/ language one may choose to utilize from both ways simultaneously. Indeed, I don't think there are any precise definitions of hard and soft analysis (yet), hence we all really make the notational abuse when writing "hard analysis" while we really mean ""hard" analysis", for instance. I believe one with a calculus background can follow the main argument Tao gives in his blog entry. The main argument is nothing but finitizing Monotone Convergence Theorem for reals (he calls it Infinite Convergence Principle). Monotone Convergence Theorem (if $\{x_n\}_n\subseteq\mathbb{R}$ is monotone and bounded then it is convergent) by itself is a soft result. Assuming the sequence is non-decreasing, normalizing it and getting rid of the limit via Cauchy Criterion, we have:

ICP: $\{x_n\}_n\subseteq[0,1]:\forall n: x_n\leq x_{n+1}$

$$\implies \forall\varepsilon,\exists N,\forall n,m\geq N: |x_n-x_m|<\varepsilon.$$

A finite version of this result would not use infinite sequences, instead it would use finite sequences. So we would need to somehow restrict the large enough indices:

Finite Convergence Principle: $\{x_n\}_n\subseteq[0,1]:\forall n: x_n\leq x_{n+1}$

$$\implies \forall\varepsilon, \forall F:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0},\forall M=M(\varepsilon,F),\exists N:1\leq N\leq N+F(N)\leq M,\forall n,m\in\{N,N+1,...,N+F(N)\}: |x_n-x_m|<\varepsilon.$$

It looks a bit messy (which is more emphasized when one writes the result as a logical statement). The main idea behind FCP is the following: Given a sequence $\{x_n\}_n\subseteq[0,1]$, instead of considering the whole sequence and its convergence, for a given error margin $\varepsilon$ we consider a finite segment of it. But we also need an arbitrary function $F$ to spars the terms of the sequence in various manners (so that we have a statement about large enough $n,m$).

Finally let us elaborate on the advantages and disadvantages of these two approaches (after Hardy's comment; admittedly this issue was what intrigued me to engage in this thread). The following points are what I have compiled from Tao's blog entry (disclaimer: the last two are due to my own understanding, and hence highly debatable):

  1. (In case you are concerned about the foundations) SA often uses the "foundational" axioms, such as AC, axiom of infinity, Dedekind Completeness Axiom, where HA one rarely does so (which is expected, everything is reduced to finite objects here).
  2. SA statements are succinct and rigorous, and they utilize precise concepts, while HA statements usually has only one of these properties: an HA statement is either "rigorous but verbose" or "succint but "fuzzy"" (compare the statements of MCT with FCP; in SA most information is swept under $\varepsilon$ and the index $N$ and the ending result says nothing more than that $N=N(\varepsilon)$, while in HA we keep track of this dependency).
  3. As HA uses weaker objects and rules in principle, it is often faster to use SA.
  4. Due to the nature of the results obtained via either way, HA is seen to be relatively easier to use in applications (e.g. when implementing a computer program about the convergence of a sequence). Indeed, SA favors existential theorems, but HA goes further than that.

One may want to take a look at this discussion at math.OF, and the following two books for a general discussion of proof mining:

Buss, S.R.. Handbook of Proof Theory. Elsevier. 1998.

Kohlenbach, Ulrich. Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Springer Monographs in Mathematics. Springer. 2008.

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