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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
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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up vote 7 down vote accepted

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

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