Tagged Questions

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Is the norm of the average $\le$ the norm of the max?

Given $\pmb X \in \mathcal{R}^p$, denote the elements of $\pmb X$ as $\pmb x_i$ for $i= 1, \dots, n$. Denote the $t(\pmb X)$ as the average of $\pmb X$ \pmb t(\pmb X) = \frac 1 n ...
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On an estimate of sequences with weights

Does there exist a $C > 0$ such that $$\sum_{n \geq 1} a_n \leq C \left( \sum_{n \geq 1} 2^n a_n^2 \right)^{1/4} \left( \sum_{n \geq 1} 2^{-n} a_n^2 \right)^{1/4}$$ for all $a_n \geq 0$ with ...
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Norm inequality? [on hold]

What are best possible constants $K_1,K_2,K_3$ would the inequality \begin{align} K_1||f||_2||g||_2 \le K_2||f||_p||g||_q \le K_3||f||_1||g||_{\infty} \end{align} holds? for all $p>1$ with ...
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Matrix Norm Inequalities and Linear Least Squares

I am working through one of my universities old QUALS and came across the following problem: Let $A,E\in \mathbb{C}^{m\times m}$. Suppose that $\sigma_\min>0$ is the smallest singular value of ...
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Is this some kind of triangle inquality?

I stumbled upon the following inequality: $$\Vert x+hz-(x+y)-(p-(x+y))\Vert_2 \geq \Vert p-(x+y)\Vert_2-\Vert x+hz-(x+y)\Vert_2$$ where $p,x,y,z \in \mathbb{R}^n$. My question is: Is this some kind ...
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Norm and Matrice Proof

I'm trying to show that the following statement is true: If $||\mathbf a - \theta \mathbf b||^2 - ||\mathbf a||^2 \geq 0$ for all $\theta \in [0,1]$, then $\mathbf a^T \mathbf b \le 0$. Is this ...
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How to prove that the spectral radius of a linear operator is the infimum over all subordinate norms of the corresponding norm of the operator.

I am trying to understand a proof I have seen of the following theorem: $$\rho(A)=\inf_{\|\cdot\|}\|A\|.$$ I understand that to do this, the idea is to show that 1) $\rho(A)\leq\|A\|$ for any norm, ...
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How to prove this inequality for operator and function

How to prove this? $\sum_{k=1}^{\infty}|(Tf)_k|^2\leq ||T||^2||f||^2$ where $T$ is an operator and a function $f$. $(Tf)_k$ is the $k$-th coordinate of Tf. Should this involve Cauchy-Schwarz or the ...
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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 $||x||_p = (\sum_{i=1}^n |x_i|^p)^{\frac{1}{p}}$ ...
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Prove relative error with condition number of matrix inequality

I was working on some questions and solutions, and encountered the following question. I am able to prove the inequality using the given information and some algebraic manipulation but the "within ...
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Inequality for norm of linear combination of linearly independent vectors

I'm trying to find a proof for the following: Let {$u_{1},...,u_{n}$} be a linearly independent set of a normed space $X$. Then, there is a constant $c>0$ such that for every set of scalars ...
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Generalization of $\frac{a + b}{c + d} \leq \text{max}(\frac{a}{c}, \frac{b}{d})$

I'm looking for a matrix version of the basic inequality for the ratio of two sums of positive numbers: $$\frac{a + b}{c + d} \leq \max\left\{\frac{a}{c}, \frac{b}{d}\right\}.$$ Specifically, I have ...
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Expression for Lp Norm [duplicate]

Use Holder inequality $(|\sum_{i=1}^n x_n\cdot y_n| \le ||x||_p\cdot ||y||_q)$ to prove that for each $x\in \Bbb R^n$: $$||x||_p=\sup_{||y||_q\le 1} {|\sum_{i=1}^n x_n\cdot y_n|}$$ tried to find for ...
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Poincare Inequality implies Equivalent Norms

I am currently working through the subject of Sobolev Spaces using the book 'Partial Differential Equations' by Lawrence Evans. After the result proving the Poincare Inequality it says the following ...
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Is it true that $2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty}$?

For any $C^1$ function defined in $(0,1)$, is it true that $$2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty}$$ If it is true, how to prove it?
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Is it true that $2^p\|f\|^p_{L_p} >2\|f\|^p_{L_\infty}$?

For any $C^1$ function defined in $(0,1)$, is it true that $$2^p\|f\|^p_{L_p} >2\|f\|^p_{L_\infty}$$ If it is true, how to prove it?
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Is $L_\infty$ norm the smallest or largest?

I am a little bit confused. For a $L_p$ function norm, is it true that for any $p<\infty$, $$\|f\|_p>\|f\|_\infty$$ Is the statement true for any domain? I want to know more inequality about ...
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Does this cross-product norm inequality hold?

Let $\times$ denote the cross-product. $\;$ Is it the case that For all unit vectors $\:\mathbf{x}\hspace{.01 in},\hspace{-0.03 in}\mathbf{y}\hspace{-0.03 in},\hspace{-0.02 in}\mathbf{z}\:$ in ...
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Prove that $||x|-|y|| \leq |x-y|$ [duplicate]

$||x|-|y|| \leq |x-y|$ when $(x,y \in R^k)$ In Principles of MA(Rudin), the author said one sees easily that $||x|-|y|| \leq |x-y|$ when $(x,y \in R^k)$ (p.88, Rudin) from the triangle ...
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Prove an inequality.

Prove that $$\displaystyle{(|x_1+y_1|^p + |x_2+y_2|^p +\dots +|x_n+y_n|^p)^{\frac{1}{p}}\leq (|x_1|^p + |x_2|^p +\dots +|x_n|^p)^{\frac{1}{p}}+(|y_1|^p + |y_2|^p +\dots +|y_n|^p)^{\frac{1}{p}}}$$ for ...
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Order of infinite dimension norms

I know that $$\|{f}\|_{L^1(0,L)}\leq\|{f}\|_{L^2(0,L)}\leq\|{f}\|_{\mathscr{C}^1(0,L)}\leq\|{f}\|_{\mathscr{C}^2(0,L)}\leq\|{f}\|_{\mathscr{C}^{\infty}(0,L)}$$ But I don't know where to put in this ...
Let $K = \left\{ v \in H_0^1(\Omega) \, : \, v \geq 0 \right\}$, further suppose $\Omega$ has the regularity property that $||v||_{H^2} \leq C(\Omega)||\Delta v||_{L^2}$, for all $v \in ... 2answers 85 views How to show for a PSD matrix$A$that$\left \| \left ( A+I \right )^{-1} \right \| \leq \frac{1}{1+\sigma _{\min}\left ( A \right )}$? If$A \in \mathbb{C}^{n \times n}$is positive semidefinite, show that$\left \| \left ( A+I \right )^{-1} \right \| \leq \frac{1}{1+\sigma _{\min}\left ( A \right )}$, where$\sigma _{\min}\left ( A ...
I am studying functional analysis and I have come across two statements which can be proven by using HÃ¶lder's inequality, but I don't know how/why. Reminder: let $a\in l^p$ and $b\in l^q$, then:  ...