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i remember in undergrad my analysis prof saying there were two types of analysis: hard analysis (sans baire category theorem) and soft analysis (using baire cat
by mlevental 8y ago
i remember in undergrad my analysis prof saying there were two types of analysis: hard analysis (sans baire category theorem) and soft analysis (using baire category theorem). i wonder if what he actually meant was in baire spaces and not in baire spaces.
- earthicus 8y agoI suspect your professor was referring to 'qualitative' vs 'quantitative' approaches when referring to soft vs hard analysis. The baire category theorem is intimately related to these to approaches, and leads to three theorems (1) the uniform boundedness principle, (2) the open mapping theorem, and (3) the closed graph theorem, that establish equivalences between the qualitative and quantitative theories. According to Tao's textbook, in practice these are not used much directly, instead one starts with quantitative bounds, then derives the qualitative corollaries. The baire category theorem tells us why those there theorems are true, and thus tells us why we can take that step from quantitative to qualitative. I think this is what your professor meant.
- mlevental 8y agocan you give an example of a qualitative corollary? do you mean a characterization of a space or something like that?
- earthicus 8y agoReferring to properties of linear operators. Examples are finiteness, surjectivity, non-negativity, monotonicity, min/max principles, contraction properties, etc.