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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

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