Talɣa:Aggur:Tusnakt/Tamyagart n ɣassa

Ɣ tusnakt, Targadda nɣ tamyagart n Cauchy–Schwarz, tettawsan ula s yism n targadda n Cauchy–Bunyakovsky–Schwarz, tga yat targadda bahra istawhman ɣ tusnakt kullut, ar sis nswurri ɣ aljibr imzirg, taslṭ, tiẓri n tsqqart d kigan n tɣawsiwin yaḍni. Targadda ad issufɣt-id yan umusnak afransis iga s yism Augustin-Louis Cauchy ɣ usggwas n 1821, sliɣ yufa Viktor Bunyakovsky yat targadda trwast akk ɣ mad izdin d aɣrd ɣ usggwas n 1859, mn bɛd yufatt daɣ yan umusnak almani Hermann Amandus Schwarz ɣ usggwas n 1888.

Mad aɣ tettini

Ar ttini targadda n Cauchy–Schwarz mas i akk sin imawayn $u$ d $v$ n yat tallunt gis afaris agnsan hann rad darnɣ tili

$|⟨𝐮,𝐯⟩{|}^2\le ⟨𝐮,𝐮⟩\cdot ⟨𝐯,𝐯⟩$

maɣ $⟨\cdot ,\cdot ⟩$ iga yan ufaris agnsan. S umdya afaris agnsan gis afaris afsnan n ilawn d ismlaln. S ɣmkad, iɣ nusi aẓur uzmir-sin ɣ tsgiwin s snat, d iɣ asn nga alugn. Ar nttafa targadda ad :

$|⟨𝐮,𝐯⟩|\le \Vert 𝐮\Vert \Vert 𝐯\Vert .$

D yat tɣawsa yaḍn, tasgiwin s snat gaddan iɣ $𝐮$ d $𝐯$ gan ilelliyn imzirgn (yaɛni gan imsadaɣn). Iɣ $u_1,\dots ,u_n\in ℂ$ d $v_1,\dots ,v_n\in ℂ$ d ufaris agnsan iga afaris agnsan asmlal anaway, hann targadda tzḍar ad ttyura zund ɣikad (maɣ tirra n taɛṛṛaḍt ar sis nmmal unaftay n ismlaln): i $𝐮,𝐯\in {ℂ}^n$, darnɣ

${|⟨𝐮,𝐯⟩|}^2={\left|\sum _{k=1}^n{u}_k{\overline{v}}_k\right|}^2\le ⟨𝐮,𝐮⟩⟨𝐯,𝐯⟩$

$=\left(\sum _{k=1}^n{u}_k{\overline{u}}_k\right)\left(\sum _{k=1}^n{v}_k{\overline{v}}_k\right)$

$=\sum _{j=1}^n|u_j{|}^2\sum _{k=1}^n|v_k{|}^2$

Ad t igan,

$|u_1{\overline{v}}_1+\cdots +u_n{\overline{v}}_n{|}^2\le (|u_1{|}^2+\cdots +|u_n{|}^2)(|v_1{|}^2+\cdots +|v_n{|}^2)$

(Ɣr uggar...)