1
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1answer
36 views

An inner product inequality

In this article: http://rgmia.org/papers/v7e/RBKIIPS.pdf, the author claims that the inequality (after (2.4)) $$\frac{|\langle a,x\rangle \langle x,b\rangle|}{\|x\|^2} \leq ...
0
votes
0answers
129 views

Prove the inequality $\sum_{i,j = 1}^n {{A_{i,j}}({x_i}^2 - {x_i}{x_j})} \ge 0 $

A is a square matrix with positive elements and x is a real vector (both of them n>1 dimensional). Prove that for any such matrix and vector $$\sum\limits_{i,j = 1}^n {{A_{i,j}}({x_i}^2 - {x_i}{x_j})} ...
0
votes
0answers
20 views

Minimum in complex inner product vector space

I'm stuck at this problem, can someone give me a hint? Let $x_i$ and $y_i$ ($i=\overline{1,n}$) be vectors in an infinite dimensional vector space $V$ with inner product $(,)$ satisfy: ...
0
votes
0answers
41 views

Apollonius’ Identity inner product space

$||z-x||^2+||z-y||^2=\frac{1}{2}||x-y||^2+2||z-\frac{x+y}{2}||^2$ I proved it by expanding both sides and i found both sides are equal. Are there any easy way to prove it?
3
votes
1answer
79 views

An inequality for inner product space: $\|x-z\|.\|y-t\|\leq \|x-y\|.\|z-t\|+\|y-z\|.\|x-t\|$

In a inner product space show that the following inequality holds. $\|x-z\|.\|y-t\|\leq \|x-y\|.\|z-t\|+\|y-z\|.\|x-t\|$ I am stuck in proving this inequality
1
vote
1answer
35 views

Prove triangle inequality of vector norm

I am trying to show that $||x+y||_p \leq ||x||_p + ||y||_p$ where $p$ is an integer larger than 1, but not infinity (I proved those cases already), and $||x||_p = (\sum_{i=1}^n |x_i|^p)^{\frac{1}{p}}$ ...
1
vote
1answer
17 views

Proving an inequality on $\sum_{1\leq i,j \leq n} \langle c_i ,c_j \rangle \times \langle l_i ,l_j \rangle$

This is a question that stumped me during an exam I took today. Let $c_1,...,c_n,l_1,...,l_n$ be vectors of $\mathbb R^n$ and $\langle .,.\rangle$ denote the dot product. Prove that ...
1
vote
1answer
91 views

Proving the inequality of Cauchy-Schwarz in an Euclidean space. [duplicate]

It says let (G, <.,.>) be an euclidean space. Show that for all x, y belonging to G: modulus<x,y> <= sqrt<x,x> * sqrt<y,y> and in the ...
0
votes
2answers
36 views

Inequality involving inner product. $|\langle u,v\rangle+\overline{\langle u,v\rangle}|\le 2|\langle u,v\rangle|$

How is the following inequality involving the inner product true? $|\langle u,v\rangle+\overline{\langle u,v\rangle}|\le 2|\langle u,v\rangle|$ I can see that for a complex number $z=u+iv, ...
0
votes
2answers
155 views

How to prove Schwarz inequality for Hermitian forms?

I'm trying to do something like the proof of the Schwarz inequality for inner product. If $h(y,y)\neq 0$, then we can take $\alpha=-h(x,y)/h(y,y)$ and calculate $h(x+\alpha y,x+\alpha y)$ which is ...
4
votes
1answer
155 views

Why doesn't Cauchy-Schwarz in $\mathbb{R}^n$ generalize to exponents $k>2$?

Given $(x_i)_{i=1}^n, (y_i)_{i=1}^n \in \mathbb{R}^n$, the Cauchy-Schwarz Inequality asserts $$\left( \sum_{i=1}^n x_i y_i \right)^2 \leq \left( \sum_{i=1}^n x_i \right)^2 \left( \sum_{i=1}^n y_i ...
4
votes
2answers
327 views

Why does the Cauchy-Schwarz inequality hold in any inner product space?

I am working through linear algebra problems in Apostol's Calculus, and he has numerous problems that seem to imply that Cauchy-Schwarz holds no matter how the inner product is defined. Then, he has ...
3
votes
1answer
113 views

How is the inner product in $H^{-1/2}$ defined?

Since $H^{1/2}$ is a Hilbert space, $H^{-1/2}$ must also be a Hilbert space by the isomorphism of Riesz representation theorem. How is the inner product defined there? We know there is a nice ...
3
votes
2answers
95 views

When does $\|x+y\|=\|x\|+\|y\|?$

Let $(V_\mathbb R,\langle,\rangle)$ be an inner product space. I'm trying to see for $x,y\in V$ when does $\|x+y\|=\|x\|+\|y\|?$ Let $\|x+y\|=\|x\|+\|y\|$ Squaring both sides, $\langle ...
0
votes
4answers
75 views

How can one prove the inequality $(|r_1s_1|+\cdots+|r_ns_n|)^2\leq(r_1^2+\cdots+r_n^2)(s_1^2+\cdots+s_n^2)$ in $\mathbb{R}^n$?

In the inner product space $\mathbb{R}^n$, Cauchy's inequality tells us that $$ (r_1s_1+\cdots+r_ns_n)^2\leq(r_1^2+\cdots+r_n^2)(s_1^2+\cdots+s_n^2). $$ Apparently the inequality can be improved to ...
2
votes
1answer
49 views

inequality in inner product

I want to show that if $$(u-\hat u, v-\hat u)\leq 0$$ and also $$(v-\hat v, u-\hat v)\leq 0$$ then $$(\hat u-\hat v, u-v)\geq 0$$ Please help me. Maybe it is easy Thanks
3
votes
1answer
136 views

cauchy schwarz inequality problemes

I have to prove that for all $x,y,z>0$, $$\left(\frac{x+y}{x+y+z}\right)^{0.5} + \left(\frac{x+z}{x+y+z}\right)^{0.5} + \left(\frac{z+y}{x+y+z}\right)^{0.5} \leq 6^{0.5}$$ using Cauchy-Schwarz ...
2
votes
2answers
237 views

Cauchy-Schwarz and Bessel's Inequalities

Deduce the Cauchy-Schwarz Inequality from the case m = 1 of Bessel’s Inequality: the sum of $$\sum_{i=1}^{m}|(v,u_i)|^2 \leq ||v||^2. $$
2
votes
1answer
39 views

Inequality in inner product space

Given $V$ an inner product space with norm $(‖v‖_V)^2$=$∫_Ω(v^2 (x)+|∇v|^2 )dx$. Prove that $$(∫_Ω(|v||w|+|∇v||∇w|)dx)^2 ≤ ∫_Ω(|v|^2+|∇v|^2 )dx ∫_Ω(|w|^2+|∇w|^2 )dx=(‖v‖_V)^2(‖w‖_V)^2.$$ Any ...
2
votes
1answer
87 views

Two inequalities related to norm

We have some difficulties in the following problem: Let $H$ be a real Hilbert space. Find $\alpha>0$ such that $$ \langle\frac{u}{\sqrt{\|u\|}}-\frac{v}{\sqrt{\|v\|}}, u-v\rangle\geq ...
3
votes
2answers
149 views

Inequality involving norm and inner product

I am stuck proving this trivial inequality: on a real inner product space, $(||x||+||y||)\frac{\langle x,y\rangle}{||x|| \cdot ||y||}\leq||x+y||$ I have tried to square both sides and use the Cauchy ...
4
votes
1answer
99 views

inequality on inner product

Let $x \in \Bbb R^n$ and $Q \in M_{n \times n}(\Bbb R)$, where $Q$ is hermitian and negative definite. Let $(\cdot,\cdot)$ be the usual euclidian inner product. I need to prove the following ...
4
votes
1answer
179 views

Cauchy-Schwarz Inequality in $\mathbb{Z}$-modules

Cauchy-Schwarz inequality for inner products If $V$ is a real vector space and $f: V\times V\to \mathbb{R}$ is a symmetric bilinear positive map, then we have the Cauchy-Schwarz inequality ...
4
votes
1answer
15k views

Proof for triangle inequality for vectors

Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below. By considering $u$ and $v$ as two sides of a ...
0
votes
1answer
173 views

Bound involving inner product

Let $a$ be a vector in $\mathbb R^n$, and let $c$ be a real number. Is there a simple characterization of the set $$\{x\in\mathbb R^n : (a,x) \geq c\}$$ where $(a,x)$ is the inner product ...
8
votes
3answers
880 views

Proofs of the Cauchy-Schwarz Inequality?

How many proofs of the Cauchy-Schwarz inequality are there? Is there some kind of reference that lists all of these proofs?