On Geometry and Sums of Squares John von Neumann đã từng nói: "Trong toán học bạn không hiểu được những điều, bạn chỉ cần có được sử dụng cho họ." Khái niệm về không gian n-chiều bây giờ là một người đăng ký đầu trong chương trình giảng dạy toán học, và vài người trong chúng ta xem nó như là đặc biệt bí ẩn, tuy nhiên, cho các thế hệ trước chúng ta đã không luôn luôn như vậy. | 4 On Geometry and Sums of Squares John von Neumann once said In mathematics you don t understand things you just get used to them. The notion of n-dimensional space is now an early entrant in the mathematical curriculum and few of us view it as particularly mysterious nevertheless for generations before ours this was not always the case. To be sure our experience with the Pythagorean theorem in R2 and R3 is easily extrapolated to suggest that for two points x x1 x2 . xd and y yi y2 . yd in Rd the distance p x y between x and y should be given by p x y ự yi - Xi 2 y2 - X2 2 ---- yd - Xd 2 but despite the familiarity of this formula it still keeps some secrets. In particular many of us may be willing to admit to some uncertainty whether it is best viewed as a theorem or as a definition. With proper preparation either point of view may be supported although the path of least resistance is surely to take the formula for p x y as the definition of the Euclidean distance in Rd. Nevertheless there is a Faustian element to this bargain. First this definition makes the Pythagorean theorem into a bland triviality and we may be saddened to see our much-proved friend treated so shabbily. Second we need to check that this definition of distance in Rd meets the minimal standards that one demands of a distance function in particular we need to check that p satisfies the so-called triangle inequality although by a bit of luck Cauchy s inequality will help us with this task. Third and finally we need to test the limits on our intuition. Our experience with R2 and R3 is a powerful guide yet it can also mislead us and one does well to develop a skeptical attitude about what is obvious and what is not. Even though it may be a bit like having dessert before having dinner 51 52 On Geometry and Sums of Squares In R2 one places a unit circle in each quadrant of the square -2 2 2. A non-overlapping circle of maximal radius is then centered at the origin. Fig. . This arrangement of 5
