متراجحة كوشي شوارتز للمتجهات في فضاء الضرب الداخلي مع التطبيقات

خالد عبد الرازق عوض الله النور

doi:10.26389/ajsrp.k210418

خالد عبد الرازق عوض الله النور

Open Access

https://doi.org/10.26389/ajsrp.k210418

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Abstract

تفردت متباينة كوشي شوارتز بالعديد من البراهين المختلفة، والتي لم يسمح المجال بذكرها جميعاً، مما أكسبها أهمية في توليد متباينات رياضية متنوعة وتقديم براهين لها. تناولت هذه الورقة براهين مختلفة لهذه النظرية في إطار فضاء الضرب الداخلي للمتجهات، مما أسهم في تأسيس متباينات رياضية مختلفة أبرزها التطبيقات التي تظهر علاقات متعددة على أضلاع المثلث .

Highlights

1- Introduction: In mathematics, the Cauchy-Schwarz inequality, is the one of the most used inequality in mathematics [1],the applications of this famous inequality include linear algebra, probability theory,as well as important topics in physics and engineering[2] .Many proofs of the Cauchy-Schwarz inequality in complex spaces are presented by Volker, see[3], Hui - hua and Shanhe introduced some proofs of in the same theorem, in the sense of series, see [4].In this paper we show some different proofs of this theorem
Applications for the generalized triangle inequality[5], KOSTADIN and RISTO obtain some new proofs of The Cauchy-Schwarz inequality for the general type of n-inner product and some applications are given[6]
Problem statement and objectives: As we know, those who concerned in mathematics faced a lot of problems and how could solve it, many students have shown that the issue of comparisons in mathematics are the most complex, that left negative results, which has prompted us to find an establish a work that would frame the solution of these sort of problems by using Cauchy- Schwarz Inequality for Vectors

Summary

Alla Elnour

0⃗ 〉 = 0 and ‖u⃗ ‖ ∙ ‖v‖= ‖u⃗ ‖ ∙ ‖0⃗ ‖ = 0, both sides of the desired inequality equal zero, so the inequality hold. V , we can apply the Figure 1pythagrean theorem inner product Pythagorean theorem for inner for product spaces, since u⃗ is the sum of u⃗⃗⃗⃗1 and p , where p ⊥ u⃗⃗⃗⃗1 , . We multiply both sides by ‖v‖2, and take the square root, we get:.

Which implies that

Equality holds exactly when