Finite-Dimensional Vector Spaces (Undergraduate Texts in

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In the later 19th Century, Georg Cantor established the first foundations of set theory, which enabled the rigorous treatment of the notion of infinity, and which has since become the common language of nearly all mathematics. Recently he has been investigating the scope of heat-flow methods and induction-on-scales arguments in the analysis of geometric inequalities arising in the restriction theory for the Fourier transform. The oldest definition of mathematical analysis as opposed to synthesis is that given in Euclid, XIII. 5, which in all probability was framed by Eudoxus: "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it."

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