proof of Cauchy-Schwarz inequality for real numbers

The version of the Cauchy-Schwartz inequality we want to prove is


where the ak and bk are real numbers, with equality holding only in the case of proportionality, ak=λbk for some real λ for all k.

The proof is by direct calculation:

k=1nak2k=1nbk2-(k=1nakbk)2 =k,l=1nak2bl2-akbkalbl

The above identity implies that the Cauchy-Schwarz inequality holds. Moreover, it is an equality only when

akbl-albk=0akbk=albl or bkak=blal or ak=bk=0,

for all k and l. In other words, equality holds only when ak=λbk for all k for some real number λ.

Title proof of Cauchy-Schwarz inequality for real numbers
Canonical name ProofOfCauchySchwarzInequalityForRealNumbers
Date of creation 2013-03-22 14:56:38
Last modified on 2013-03-22 14:56:38
Owner stitch (17269)
Last modified by stitch (17269)
Numerical id 5
Author stitch (17269)
Entry type Proof
Classification msc 15A63