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University-数学讨论区-Linear Algebra, Advanced Algebra
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发表于 11-8-2009 07:37 PM
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发表于 4-11-2009 02:18 AM
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let z=x+iy
prove that |z| < |x| + |y| < √2 |z|
这题要怎样prove呢？ |
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发表于 4-11-2009 08:52 PM
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原帖由 eric336 于 4-11-2009 02:18 AM 发表 
let z=x+iy
prove that |z| < |x| + |y| < √2 |z|
这题要怎样prove呢？
|z| = |x + iy| =< |x| + |iy| = |x| + |y| (triangle inequality)
√2 |z| = √[2 (x^2 + y^2) ] >= √(x^2 + y^2 + 2|xy|) = √(|x|+|y|)^2 = |x| + |y| ( x^2 + y^2 >= 2|xy|) |
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