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Saturday, January 24, 2015

Cauchy–Schwarz

Finally I understand the Cauchy–Schwarz inequality! No induction needed.

Theorem
a,bRatn
nNat
                     
(ab)2Q(a)Q(b)

Proof
Recall that by definition
ab=ni=1aibiQ(a)=ni=1a2iQ(b)=ni=1b2i.
The claim, reformulated as
(ni=1aibi)2(ni=1a2i)(ni=1b2i),
is equivalent to (by distributing terms)
(ni=1a2ib2i)+1i<jn2aibiajbj(ni=1a2ib2i)+1i<jna2ib2j+a2jb2i
and (by collecting terms)
01i<jna2ib2j+a2jb2i2aibiajbj=1i<jn(aibjajbi)2.
The last statement is true because squares are positive and sums of positive numbers are positive.

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