What is the difference between hard and soft analysis?

A historical note.

Hermann Weyl mentioned in his talk "Felix Kleins Stellung in der mathematischen Gegenwart" that the dichotomy of "hard vs. soft analysis" had been suggested by Hardy. According to Hardy, there is the function theory of the "hard, sharp, narrow" kind (due to Bohr, Landau or Littlewood) as opposed to the "soft, large, vague" kind (due to Birkhoff or Koebe).

Edit. Apparently, Hardy's musings are contained in his paper "Prolegomena To a Chapter on Inequalities" (unfortunately, I don't have access to it at the moment).

Edit 2. Indeed, here's the quotation from Hardy's paper.

A thorough mastery of elementary inequalities is to-day one of the first necessary qualifications for research in the theory of functions; at any rate, in function theory of the "hard, sharp, narrow" kind as opposed to the "soft, large, vague" kind (I do not use any of these adjectives as words either of praise or blame), the function-theory of Bohr, Landau, or Littlewood, as opposed to the function-theory of Birkhoff or Koebe. It is essential to anyone working in this field to be master both of the main results and of the tricks of the trade.

Disclaimer: I'm no expert-this is really a question for the analysts and historians of mathematics.

As far as I know,the terminology came into existence in the early 20th century distinguish the classical "calculus" type analysis (hard analysis) from the new point set topology/functional analysis approach (soft analysis). A hard analytic argument uses a direct calculation or construction of an exact estimate bounds of specific function or function types to prove a statement. A soft analytic arguement uses the general topological or geometric properties of a space in which a function or function class is defined to prove a result indirectly without a precisely calculated "bound".

For example, the fact that the Cantor set has measure zero is a "hard" analytic arguement; it uses an epsilon-delta arguement to show the limit of the sequence of "slices" of the lengths of it's component intervals on the real line converges to 0.

An example of a "soft" analytic arguement: (IVT) Let $f$ be a continuous function defined on a connected subset of the real line i.e. an interval with a well defined least upper bound and greatest lower bound. Then the function is defined at every point inbetween the lub and the glb. A soft proof would be as follows: Since an interval $I$ of $\Bbb R$ is a connected subset of $\Bbb R$ and $f$ is continuous, then $f(I)$ is also connected. Therefore, for every $x \in I$, $f(x)$ is in $f(I)$. Notice this proof does not involve a direct computation of bounds that proves $f(x)$ is in the image set of $f$ (although it certainly COULD be proven that way).

Anyway, that's how Gerald Itzkowitz taught it to me and I learned a long time ago to trust him on these matters........

Consider the problem of shuffling a deck of cards using some shuffling technique. One may wonder, "If I use this or that shuffling technique, will I shuffle the deck?" The problem can be transferred to a question of whether a particular finite Markov process converges to the uniform distribution or not. Omitting some details, a classical theorem says that, yes, the process will converge (to the uniform distribution) as long as your technique is reasonable. Not only that, the convergence will eventually be exponential. This seems like a useful theorem, but it is actually rather deficient. The problem is that your practical side may wonder how many shuffles are required to get the deck reasonably randomized, and this theorem doesn't help. So you might say that the original analysis was soft because the result does not help in solving this quantitative problem. It tells you that shuffling is a good idea, but it doesn't give you any clue whether a given technique could shuffle the deck in your lifetime or not. A hard analysis would tell you, for example, "If one defines reasonably randomized by measure blahblah, then $2\log_2(52)$ riffle shuffles are sufficient to randomize the deck."