# cauchyschwarz

1. ### Cauchy-Schwarz inequality (pre-inner product)

Greet you, 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...
2. ### Cauchy-Schwarz inequality

Hi, I am stuck on this question. Prove |f(L)-f(0)|^2 \le L \int_0^L |f'(x)|^2 dx for any function f \in C^1([0,L]). So I know I should use the Cauchy-Schwarz inequality applied to the functions f′ and 1, but I am stuck on how to do it. Thanks for any help people.
3. ### Cauchy-Schwarz Inequality Proof

So I'm reading this document that has a proof of the Cauchy-Schwarz Inequality, and I'm missing the logic. It says, "By Proposition 2.3 and (i) of Proposition 2.5, we have (assuming y \ne 0, otherwise, nothing needs to be proved)" (The propositions referenced basically establish the linearity...
4. ### Cauchy-Schwarz inequality

Hi, one of my books states the inequality as |x^*y| \leq ||x|| ||y|| for all x,y\in\mathbb{C}^{n\times 1}, and says that equality holds if and only if y=\alpha x for \alpha =x^*y/x^*x. To me it looks like equality holds if y=\alpha x for any \alpha\in\mathbb{C}...
5. ### Applications of Cauchy-Schwarz

Completely lost on this one, how do I even begin?
6. ### Cauchy-Schwarz Inequality

I am supposed to prove that for all real numbers a,b, \theta that (acos\theta+bsin\theta)^2\leq a^2+b^2 I've only gotten up to this point which is where I am not sure as to how to appply the cauchy schwarz inequality, (acos\theta+bsin\theta)^2=a^2+b^2-(asin\theta+bcos\theta)^2 Am I on the...
7. ### Cauchy-Schwarz Inequality (complex analysis)

Nothing makes me stare off into space and wonder why I'm bothering with mathematics at all more than a proof presented with the intent of educating the reader yet the opening statement of which lacks substantive intuitive relevance. So, the proof I want to understand be damned, I come asking for...
8. ### Cauchy-Schwarz inequality for Integrals

Suppose that the functions f,g,f^2,g^2, fg are integrable on the closed, bounded interval [a,b]. Prove that: \int ^b _a fg \leq \sqrt { \int ^b_a f^2 } \sqrt { \int ^b_a g^2 } . Proof. Now, 0 \leq \sqrt { \int _a ^b (f- \lambda g)(f- \lambda g) } ^2 = \int ^b_a (f^2- \lambda fg -...
9. ### CauchySchwarz Inequality Help :D

||x||=4, ||y||=5,<x,y>=8. What is the cauchyshwarz inequality? <This was relatively straight forward.. Not too sure about the rest however.. What is the norm of 4x-4y? What is the cosine angle between x and 4x-4y? Find λ, such that 2x-y and λx-y are orthogonal? Many thanks
10. ### cauchy-schwarz problem

by considering suitable vectors in R4, show that only one choice of real numbers x,y,z satisfies : 3(x^2+y^2+z^2+4) - 2( yz+zx+xy) -4(x+y+z)=0 and find these numbers
11. ### Cauchy-Schwarz Inequality

If a_{1}, \ldots, a_{n} and b_{1} \ldots, b_{n} are arbitrary real numbers, then \left(\sum_{k=1}^{n} a_{k}b_{k} \right)^{2} \leq \left(\sum_{k=1}^{n} a_{k}^{2} \right) \left(\sum_{k=1}^{n} b_{k}^{2} \right) . If some a_{i} \neq 0 then equality holds if and only if there is a real x...