What is Cauchy-schwarz inequality: Definition and 26 Discussions

In mathematics, the Cauchy–Schwarz inequality, also known as the Cauchy–Bunyakovsky–Schwarz inequality, is a useful inequality in many mathematical fields, such as linear algebra, analysis, probability theory, vector algebra and other areas. It is considered to be one of the most important inequalities in all of mathematics.The inequality for sums was published by Augustin-Louis Cauchy (1821), while the corresponding inequality for integrals was first proved by
Viktor Bunyakovsky (1859). The modern proof of the integral version was given by Hermann Schwarz (1888).

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  1. C

    I Derivation of Cauchy-Schwarz Inequality

    For this, I don't understand how they got from (1) to (2)? Dose someone please know what binary operation allows for that? I also don't understand how they algebraically got from line (2) to (3). Many thanks!
  2. P

    Evaluating the Integral of a Vector Field Using Cauchy-Schwarz Inequality

    Here is my attempt (Note: ## \left| \int_{C} f \left( z \right) \, dz \right| \leq \left| \int_C udx -vdy +ivdx +iudy \right|## ##= \left| \int_{C} \left( u+iv, -v +iu \right) \cdot \left(dx, dy \right) \right| ## Here I am going to surround the above expression with another set of...
  3. Rabindranath

    A Lagrange multipliers on Banach spaces (in Dirac notation)

    I want to prove Cauchy–Schwarz' inequality, in Dirac notation, ##\left<\psi\middle|\psi\right> \left<\phi\middle|\phi\right> \geq \left|\left<\psi\middle|\phi\right>\right|^2##, using the Lagrange multiplier method, minimizing ##\left|\left<\psi\middle|\phi\right>\right|^2## subject to the...
  4. J

    MHB Understanding the Cauchy-Schwarz Inequality

    I am not sure what examples to give, need help on this. Have attached the theorem as well.
  5. E

    I Spivak's proof of Cauchy Schwarz

    I was browsing through Spivak's Calculus book and found in a problem a very simple way to prove the cauchy schwarz inequality. Basically he tells to substitute x=xᵢ/[√(x₁²+x₂²)] and similarly for y (i=1 and 2), put into x^2 + y^2 >= 2xy. Add the two cases and we get the result. The problem is...
  6. L

    Inner Product, Triangle and Cauchy Schwarz Inequalities

    Homework Statement Homework Equations I am not sure. I have not seen the triangle inequality for inner products, nor the Cauchy-Schwarz Inequality for the inner product. The only thing that my lecture notes and textbook show is the axioms for general inner products, the definition of norm...
  7. Math Amateur

    MHB Cauchy-Schwarz Inequality - Duistermaat and Kolk, CH. 1, page 4 .... ....

    I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of the Cauchy-Schwarz Inequality ... Duistermaat and Kolk"s proof of the Cauchy-Schwarz Inequality...
  8. Mr Davis 97

    I Understanding Cauchy-Schwarz Inequality

    I am trying to find the max and min values of the function ##f(x,y) = 2\sin x \sin y + 3\sin x \cos y + 6 \cos x##. By the Cauchy-Schwarz inequality, we have that ##|f(x,y)|^2 \le (4+9+36) (\sin^2 x \sin^2y + \sin^2 x \cos^2 y + \cos^2 x) = 49##. Hence ##-7 \le f(x,y) \le 7##. My question has...
  9. Y

    What is the discriminant of the following quadratic equation

    quadratic equation ||v||^2 - c(2v·w)+c^2||w||^2=0, where c belongs to any real number, v and w are both vectors
  10. H

    I Validity of proof of Cauchy-Schwarz inequality

    Proof: If either x or y is zero, then the inequality |x · y| ≤ | x | | y | is trivially correct because both sides are zero. If neither x nor y is zero, then by x · y = | x | | y | cos θ, |x · y|=| x | | y | cos θ | ≤ | x | | y | since -1 ≤ cos θ ≤ 1 How valid is this a proof of the...
  11. mnb96

    A Question on Cauchy-Schwarz inequality

    Hello, if we consider the vector spaces of integrable real functions on [a,b] with the inner product defined as: \left \langle f,g \right \rangle=\int _a^bf(x)g(x)dx the Cauchy-Schwarz inequality can be written as: \left | \int_{a}^{b} f(x)g(x)dx\right | \leq \sqrt{\int_{a}^{b}f(x)^ 2dx}...
  12. R

    MHB Show that equality holds in Cauchy-Schwarz inequality if and only if....

    This is from section I 4.9 of Apostol's Calculus Volume 1. The book states the Cauchy-Schwarz inequality as follows: \left(\sum_{k=1}^na_kb_k\right)^2\leq\left(\sum_{k=1}^na_k^2\right)\left(\sum_{k=1}^nb_k^2\right) Then it asks you to show that equality holds in the above if and only if there...
  13. M

    MHB Cauchy-Schwarz inequality for pre-inner product

    Dear all, I've encountered some problems while looking through the book called "Operator Algebras" by Bruce Blackadar. At the very beginning there is a definition of pre-inner product on the complex vector space: briefly, it's the same as the inner product, but the necessity of x=0 when [x,x]=0...
  14. phosgene

    Valid proof of Cauchy-Schwarz inequality?

    Homework Statement I was discussing the proof for the Cauchy-Schwarz inequality used in our lectures, and another student suggested an easier way of doing it. It's really, really simple. But I haven't seen it anywhere online or in textbooks, so I'm wondering if it's either wrong or is only...
  15. A

    Equality in the Cauchy-Schwarz inequality for integrals

    Homework Statement Regarding problem 1-6 in Spivak's Calculus on Manifolds: Let f and g be integrable on [a,b]. Prove that |\int_a^b fg| ≤ (\int_a^b f^2)^\frac{1}{2}(\int_a^b g^2)^\frac{1}{2}. Hint: Consider seperately the cases 0=\int_a^b (f-λg)^2 for some λ\inℝ and 0 < \int_a^b (f-λg)^2 for...
  16. S

    Using the Cauchy-Schwarz inequality to prove all real values for a, b, and theta

    Homework Statement Use the Cauchy-Schwarz inequality to prove that for all real values of a, b, and theta (which ill denote as θ), (a cosθ + b sinθ)2 ≤ a2 + b2 Homework Equations so the Cauchy-Schwarz inequality is | < u,v>| ≤ ||u|| ||v|| The Attempt at a Solution I'm having...
  17. S

    How Does the Cauchy-Schwarz Inequality Prove a Vector Inequality?

    Homework Statement Let u = [a b] and v = [1 1]. Use the Cauchy-Schwarz inequality to show that (a+b/2)2 ≤ a2+b2/2. Those vectors are supposed to be in column form. Homework Equations |<u,v>| ≤||u|| ||v||, and the fact that inner product here is defined by dot product (so <u,v> = u\cdotv)...
  18. L

    How can the Cauchy-Schwarz inequality be applied to L^2 functions in a book?

    Hi, Quick question here: I know that C-S inequality in general states that |<x,y>| \leq \sqrt{<x,x>} \cdot \sqrt{<y,y>} and, in the case of L^2(a,b)functions (or L^2(R) functions, for that matter), this translates to |\int^{b}_{a}f(x)g(x)dx| \leq \sqrt{\int^{b}_{a}|f(x)|^2dx} \cdot...
  19. V

    Proof of Cauchy-Schwarz Inequality

    Homework Statement Let V be a vector space with inner product <x,y> and norm ||x|| = <x,x>^1/2. Prove the Cauchy-Schwarz inequality <x,y> <= ||x|| ||y||. Hint given in book: If x,y != 0, set c = 1/||x|| and d = 1/||y|| and use the fact that ||cx ± dy|| >= 0. Here...
  20. K

    Cauchy-Schwarz inequality: cov(X,Y)]^2 ≤ var(X) var(Y)

    Suppose that X andy Y are (scalar) random variables. Show that [cov(X,Y)]^2 ≤ var(X) var(Y). (Cauchy-Schwarz inequality) Sow that equality holds if and only if there is a relationship of the form m.s. c=aX+bY (i.e. c is equal to aX+bY in "mean square"). =========================...
  21. E

    What Is the Correct Form of the Cauchy-Schwarz Inequality?

    Hello, For two n-dimensional vectors \mathbf{v}_1\text{ and }\mathbf{v}_2, what is the Cauchy-Schwarz Inequality: 1- |\mathbf{v}_1.\mathbf{v}_2|\leq\|\mathbf{v}_1\|\|\mathbf{v}_2\|, or 2- |\mathbf{v}_1.\mathbf{v}_2|\leq\|\mathbf{v}_1\|+\|\mathbf{v}_2\| In either case, the equality...
  22. H

    When does equality hold in Cauchy-Schwarz inequality

    Homework Statement Prove that if V is a vector space over \mathbb{C}^n with the standard inner product, then |<x,y>| = ||x|| \cdot ||y|| implies one of the vectors x or y is a multiple of the other. The Attempt at a Solution Assume the identity holds and that y is not zero...
  23. Fredrik

    Proving Cauchy-Schwarz Inequality: My Dumb Question

    I have to ask a dumb question. I seem to be doing something very wrong here, and it's probably trivial, but for some reason I don't see what it is. I decided to try to prove the Cauchy-Schwarz inequality without opening a book. I remember that a proof I read once started by noting that...
  24. M

    Cauchy-Schwarz Inequality Proof Question

    Homework Statement Prove the Cauchy-Schwarz Inequality. Homework Equations |\mathbf{x \cdot y}| \leq |\mathbf{x}||\mathbf{y}|, \forall \mathbf{x,y} \in \mathbb{R}^{n} (1) The Attempt at a Solution If x is equal to 0, then both sides are equal to 0. If x not equal to 0 the following...
  25. B

    Proof of Cauchy-Schwarz Inequality

    I have a homework problem in which I have to prove the Cauchy-Schwarz inequality. I tried to do it by induction, but when I try to do summation to 2, I get a mess of terms. The professor hinted that one can use the fact that geometric means are less than or equal the arithmetic mean, but I can't...
  26. T

    The Cauchy-Schwarz inequality for real numbers

    Ok...here is some back ground into my new found situation. I have done very well in every math class up to this point in time so I felt it was time for me to start looking at taking some more difficult classes. That being said I am technically still in my freshman year in college so I may have...