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The Cauchy-Schwarz Inequality

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The Cauchy-Schwarz Inequality The Cauchy-Schwarz Inequality Proofs and applications in various spaces Cauchy-Schwarz olikhet Bevis och tillämpningar i olika rum Thomas Wigren Faculty of Technology and Science Mathematics, Bachelor Degree Project 15.0 ECTS Credits Supervisor: Prof. Mohammad Sal Moslehian Examiner: Niclas Bernhoff October 2015 THE CAUCHY-SCHWARZ INEQUALITY THOMAS WIGREN Abstract. We give some background information about the Cauchy-Schwarz inequality including its history. We then continue by providing a number of proofs for the inequality in its classical form using various proof tech- niques, including proofs without words. Next we build up the theory of inner product spaces from metric and normed spaces and show applications of the Cauchy-Schwarz inequality in each content, including the triangle inequality, Minkowski's inequality and H¨older'sinequality. In the final part we present a few problems with solutions, some proved by the author and some by others. 2010 Mathematics Subject Classification. 26D15. Key words and phrases. Cauchy-Schwarz inequality, mathematical induction, triangle in- equality, Pythagorean theorem, arithmetic-geometric means inequality, inner product space. 1 2 THOMAS WIGREN 1. Table of contents Contents 1. Table of contents2 2. Introduction3 3. Historical perspectives4 4. Some proofs of the C-S inequality5 4.1. C-S inequality for real numbers5 4.2. C-S inequality for complex numbers 14 4.3. Proofs without words 15 5. C-S inequality in various spaces 19 6. Problems involving the C-S inequality 29 References 34 CAUCHY-SCHWARZ INEQUALITY 3 2. Introduction The Cauchy-Schwarz inequality may be regarded as one of the most impor- tant inequalities in mathematics. It has many names in the literature: Cauchy- Schwarz, Schwarz, and Cauchy-Bunyakovsky-Schwarz inequality. The reason for this inconsistency is mainly because it developed over time and by many people. This inequality has not only many names, but also it has many manifestations. In fact, the inequalities below are all based on the same inequality. n 2 n n P P 2 P 2 (1) aibi ≤ ai bi . i=1 i=1 i=1 2 (2) R 1 f (x) g (x)dx ≤ R 1 jf (x)j2dx R 1 jg (x)j2dx. 0 0 0 (3) jhu; vij ≤ kuk kvk. This versatility of its forms makes it a well-used tool in mathematics and it has many applications in a wide variety of fields, e.g., classical and modern analysis including partial differential equations and multivariable calculus as well as ge- ometry, linear algebra, and probability theory. Outside of mathematics it finds its usefulness in, for example, physics, where it plays a role in the uncertainty relations of Schr¨odingerand Heisenberg [1]. Recently, there have been several extensions and generalizations of the C-S inequality and its reverse in various settings such as matrices, operators, and C∗-algebras, see e.g., [6, 14]. These topics are of special interest in operator theory and matrix analysis; cf. [1,4]. Understanding the classical results on C-S inequality is definitely a starting point for doing some research in these research areas. This thesis is organized as follows. In chapter 3, we start up with a brief history of the C-S inequality. Chapter 4 is devoted to classical proofs of the inequality, both in real and complex case, with some additional proofs that don't use any words. In chapter 5, we look into some spaces and show different applications of the inequality. Finally, in chapter 6, we provide a few examples of problems with various difficulties. 4 THOMAS WIGREN 3. Historical perspectives The Cauchy-Schwarz (C-S) inequality made its first appearance in the work Cours d'analyse de l'Ecole´ Royal Polytechnique by the French mathematician Augustin-Louis Cauchy (1789-1857). In this work, which was published in 1821, he introduced the inequality in the form of finite sums, although it was only writ- ten as a note. In 1859 a Russian former student of Cauchy, Viktor Yakovlevich Bunyakovsky (1804-1889), published a work on inequalities in the journal M´emoires de l'Acad´emie Imp´erialedes Sciences de St-P´etersbourg. Here he proved the inequality for infi- nite sums, written as integrals, for the first time. Figure 1. (From left) Augustin-Louis Cauchy, Viktor Yakovlevich Bunyakovsky and Karl Hermann Amandus Schwarz. In 1888, Karl Hermann Amandus Schwarz (1843-1921) published a work on minimal surfaces named Uber¨ ein die fl¨achenkleinsten fl¨acheninhaltsbetreffendes problem der variationsrechnung in which he found himself in need of the integral form of Cauchy's inequality, but since he was unaware of the work of Bunyakovsky, he presented the proof as his own. The proofs of Bunyakovsky and Schwarz are not similar and Schwarz's proof is therefore considered independent, although of a later date. A big difference in the methods of Bunyakovsky and Schwarz was in the rigidity of the limiting process, which was of bigger importance for Schwarz. The arguments of Schwarz can also be used in more general settings like in the framework of inner product spaces [19]. CAUCHY-SCHWARZ INEQUALITY 5 4. Some proofs of the C-S inequality There are many ways to prove the C-S inequality. We will begin by looking at a few proofs, both for real and complex cases, which demonstrates the validity of this classical form. Most of the following proofs are from H.-H Wu and S. Wu [24]. We will also look at a few proofs without words for the inequality in the plane. Later on, when we've established a more modern view of the inequality, additional proofs will be given. 4.1. C-S inequality for real numbers. Theorem 4.1. If a1; : : : ; an and b1; : : : ; bn are real numbers, then n !2 n n X X 2 X 2 aibi ≤ ai bi ; (4.1) i=1 i=1 i=1 or, equivalently, v v n u n u n X uX 2uX 2 aibi ≤ t ai t bi : (4.2) i=1 i=1 i=1 First proof [24]. We will use mathematical induction as a method for the proof. First we observe that 2 (a1b2 − a2b1) ≥ 0: By expanding the square we get 2 2 (a1b2) + (a2b1) − 2a1b2a2b1 ≥ 0: After rearranging it further and completing the square on the left-hand side, we get 2 2 2 2 2 2 2 2 2 2 2 2 a1b1 + 2a1b1a2b2 + a2b2 ≤ a1b1 + a1b2 + a2b1 + a2b2 or, equivalently, 2 2 2 2 2 (a1b1 + a2b2) ≤ a1 + a2 b1 + b2 : By taking the square roots of both sides, we reach q q 2 2 2 2 ja1b1 + a2b2j ≤ a1 + a2 b1 + b2; (4.3) 6 THOMAS WIGREN which proves the inequality (4.2) for n = 2. Assume that inequality (4.2) is true for any n terms. For n + 1, we have that v v v v un+1 un+1 u n u n uX 2uX 2 uX 2 2 uX 2 2 t ai t bi = t ai + an+1t bi + bn+1: (4.4) i=1 i=1 i=1 i=1 By comparing the right-hand side of equation (4.4) with the right-hand side of inequality (4.3) we know that v v v v u n u n u n u n uX 2 2 uX 2 2 uX 2uX 2 t ai + an+1t bi + bn+1 ≥ t ai t bi + jan+1bn+1j : i=1 i=1 i=1 i=1 Since we assume that inequality (4.2) is true for n terms, we have that v v u n u n n uX 2uX 2 X t ai t bi + jan+1bn+1j ≥ aibi + jan+1bn+1j i=1 i=1 i=1 n+1 X ≥ aibi; i=1 which proves the C-S inequality. Second proof [24]. We will use proof by induction again, but this time we deal with sequences. Let n !2 n n X X 2 X 2 Sn = aibi − ai bi : i=1 i=1 i=1 CAUCHY-SCHWARZ INEQUALITY 7 We have Sn+1 − Sn = n !2 n ! n ! X X 2 2 X 2 2 = aibi + an+1bn+1 − ai + an+1 bi + bn+1 i=1 i=1 i=1 n !2 n ! n ! n !2 n n X X 2 2 X 2 2 X X 2 X 2 = aibi + an+1bn+1 − ai + an+1 bi + bn+1 − aibi + ai bi i=1 i=1 i=1 i=1 i=1 i=1 n n n 2 X 2 2 X 2 X = −bn+1 ai − an+1 bi + 2an+1bn+1 aibi i=1 i=1 i=1 2 2 2 2 2 2 2 2 = − bn+1 a1 + a2 + ··· + an + an+1 b1 + b2 + ··· + bn − 2an+1bn+1 (a1b1 + a2b2 + ··· + anbn) 2 2 2 = − (bn+1a1 − an+1b1) + (bn+1a2 − an+1b2) + ··· + (bn+1an − an+1bn) n X 2 = − (bn+1ai − an+1bi) ; i=1 from which we get Sn+1 ≤ Sn. Hence Sn ≤ Sn−1 ≤ ::: ≤ S1 = 0, which yields the C-S inequality. Third proof [24]. We start this proof by observing that the inequality n n 1 X X 2 (a b − a b ) ≥ 0 (4.5) 2 i j j i i=1 j=1 is always true. Expanding the square and separating the sums give us n n 1 X X 2 0 ≤ (a b − a b ) 2 i j j i i=1 j=1 n n 1 X X = a2b2 + a2b2 − 2a b a b 2 i j j i i i j j i=1 j=1 n n n n n n 1 X X 1 X X X X = a2b2 + a2b2 − a b a b 2 i j 2 j i i i j j i=1 j=1 j=1 i=1 i=1 j=1 n n n n n n 1 X X 1 X X X X = a2 b2 + a2 b2 − a b a b : 2 i j 2 j i i i j j i=1 j=1 j=1 i=1 i=1 j=1 Renaming the indices and reshaping the formula give us n n n n n n 1 X X 1 X X X X a2 b2 + a2 b2 − a b a b ≥ 0; 2 i i 2 i i i i i i i=1 i=1 i=1 i=1 i=1 i=1 8 THOMAS WIGREN from where n n n !2 X 2 X 2 X ai bi − aibi ≥ 0; i=1 i=1 i=1 which yields the C-S inequality.
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