Questions tagged [cauchy-schwarz-inequality]

Problems with using C-S (Cauchy-Schwarz inequality)

644 questions
25 views

Proving Cauchy-Schwartz inequality without the vanish assumption of inner products

I've came accross the following question in a book that I'm studying from, about Hilbert spaces - And the answer is - What I don't understand is why implies that - ?
48 views

Cauchy-Schwarz Inequality question

Find the number of ordered quadruples $(a,b,c,d)$ of nonnegative real numbers such that \begin{align*} a^2 + b^2 + c^2 + d^2 &= 4, \\ (a + b + c + d)(a^3 + b^3 + c^3 + d^3) &= 16. \end{align*} ...
90 views

Let $n_1,\ldots,n_k$ be positive integers summing to $N$. What's an upper bound for $\sum_{i=1}^k1/\sqrt{n_i}$?

Disclaimer. Sorry, I haven't looked into this one in any detail (as I should have). I was just thinking there out-of-be an elementary principle out here (pigeon-hole, Cauchy-Schwarz, Jensen, etc.). ...
82 views

Prove that $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$

I am trying to prove the above statement but I'm not sure if my proof is correct. My proof is as follows, Given $u\cdot v$, we know by the C-E Inequality that $|u \cdot v| \leq ||u|| \ ||v||$ ...
89 views

Prove that $\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$ when $f(f(x))=x^2$

Let $f:[0, \infty) \to [0,\infty)$ be a differentiable function with $f'$ continuous. If $f(f(x))=x^2$, prove that $$\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$$ without explicitly finding $f.$ Since we ...
105 views

49 views

How to show following inequality using the Cauchy-Schwarz Inequality?

$A$ is $M_{n\times n}$ matrix and $x,y$ are column vector. I want to show that $||A(x-y)||_2\leq ||A||_2||x-y||_2$ with the Euclidean norm. I know that for the Cauchy-Schwarz inequality, both vectors ...
135 views

Confusing Cauchy-Schwarz Inequality Proof

Teacher proved it like this: Very elegant (way simpler than most of the ones I find online), but I'm still not convinced -- particularly the last two steps or so, where the absolute value on the left-...
93 views

Show that equality holds in the Cauchy-Schwarz inequality |⟨x, y⟩| ≤ ∥x∥ ∥y∥ . for x, y if and only if x and y are linearly dependent…

Could someone please help me with this proof? I have written up this but I not sure if it is a full proof, since it is an if and only if statement. Could someone read and inform me where I can go ...
76 views

87 views

If $a+b+c=1$ and a,b,c ＞0 prove $\dfrac{b^2}{a+b^2}+\dfrac{c^2}{b+c^2}+\dfrac{a^2}{c+a^2} \geqslant \dfrac{3}{4}$ [duplicate]

If $a+b+c=1$ and a,b,c＞0 prove $\dfrac{b^2}{a+b^2}+\dfrac{c^2}{b+c^2}+\dfrac{a^2}{c+a^2} \geqslant \dfrac{3}{4}$. I tried with CS Engel form,homogenization but ina anyway i can't prove inequality. Can ...
Background: These are super boring questions but I'm trying to learn about CS inequality for probability... any help would be greatly appreciated. Thank you. Say $x = (1,2)$ and $y = (3,4)$ then ...