定理对x=(x,x2,…,x,)eC”C"→R分别定义三个函数 k 1一范数： 2=(x,)月 2-范数（或Euclid范数） xl。=max,l ∞一范数（或最大值范数）。 它们均构成范数。 说明：在同一个向量空间，可以定义多种向量范数，而对 于同一个向量，不同定义的范数，其大小可能不同。 x=1,2.-3y,=6L,=4=3
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引理2.1.1如果实数p21,g21,+-1 p g 则对于任意非负实数a,b,成立b≤a+ p g 证若ab-0,显然结论成立。下面只讨论a>0且b>0的情况。 考虑函数 1P 1-9 0(t)= (0<t<+0) p g tP+9-1 0'(t)= p()≥p(1)=1(0<1<+0) 11 令t=a9bp 即证 Is pro ced by trial v f Print2Flash Visit
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引理2.1.2(H6lder不等式) 如果实数p≥1g≥1+1-1则对于任意数组 p q a=(a1,a2,,an),b=61,b2,…,bn） 成立a,bsa,r运b
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证：令a,w=包 ,其中 代入上述不等式，则有 m r-） las mn p m q nd 交asm2ar+2s” m2立a2 Iby trial of Print2Flash
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Minkowski不等式：设 a=[a,a,…,a],b=[b,b,…,bn]∈C" 则对任何p之1都有 ②a+门2+a
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