Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.

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convergence in different normed spaces

In my class lecture notes, there is such a lemma. Let $X$ be a vector space over $\mathbb F$ and $\lVert \cdot \rVert_1$, $\lVert \cdot \rVert_2$ be two norms on it. If there is $M > 0$ such that ...
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32 views

Maximum of norm

Given a matrix $A$ with $N$ rows and $d$ columns, I would like to prove (or disprove) the following. Let $q(f)=\|(\begin{pmatrix} ...
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46 views

What kind of matrix norm satisifies $\text {norm} (A*B)\leq \text {norm} (A)*\text {norm} (B)$ in which A is square?

$||A\times B||\le ||A||\cdot ||B||$ is not always correct. But which kind of matrix norm satisifies this formula for square matrix $A$ and arbitrary matrix $B$?
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69 views

Holder Inequality when $0 < p < 1$ [duplicate]

If $0 < p < 1$, $f \in L^p$, and $\int \lvert g \rvert^q < \infty$, show that $$\int \lvert fg \rvert \ge (\int \lvert f \rvert^p)^{\frac{1}{p}}(\int \lvert g \rvert^q)^{\frac{1}{q}}$$ My ...
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65 views

Expected value of norm of multivariate normal distribution random vector

Let $X$ is a random vector size $p$ from multivariate normal distribution $\mathcal{N}$($0$, $\sigma$ $I$), $I$ is identity matrix. I want to find the expected value of reciprocal of norm like this ...
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27 views

“Transference” Argument

In the proof of the Iwaniec-Martin theorem (giving a bound in $L^p$ for the Riesz transform, $\|R_j\|_p=\cot(\frac{\pi}{2p^*})$ the proof of this equality is given by proving the inequalities $\leq$ ...
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14 views

condition to be a norm on a finite dimensional complex vector space

Here x is an element of C^n, where n is a natural number and C is the complex number field, and p is any positive number. This becomes a norm when p>=1. But, what happens when p is between 0 and 1? ...
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19 views

Order of two vectors to maximise the norm

Given vectors ${\bf a} = [a_1, \dots , a_n]^T$ and ${\bf b} = [b_1, \dots , b_n]^T$, a permutation $\pi$ acting on $[1, \dots ,n]$ and defining ${\bf b}^{\pi} = [b_{\pi(1)}, \dots , b_{\pi(n)}]^T$, ...
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28 views

Is the countable product of normed spaces a normed space with this “norm”

Consider the product $\mathcal{N} = \mathcal{N}_{1} \times \mathcal{N}_{2} \times \mathcal{N}_{3} \times \dots$ where $\mathcal{N}_{1}, \mathcal{N}_{2}, \mathcal{N}_{3}, \dots$ are normed spaces with ...
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20 views

Parallelogram law using complex inner product not adding up

Does the parallelogram law still hold in the complex case? Using the following definitions: $\langle \textbf{x}, \textbf{y} + \textbf{z} \rangle = \langle \textbf{x}, \textbf{y} \rangle + \langle ...
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31 views

Banach space and its closed subspace. a vectors satisfying inequality.

V=a Banach space over R W=a proper closed subspace of V Prove : For any $\epsilon > 0$, there is a v $\in$_V_ such that ||v||=1 and ||v+w||$\geq$ $1$ - $\epsilon$ I have shown that there exists ...
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26 views

vector norms involving quantities with a filter relationship

I have two vector norm quantities: $\|\Psi^T(t)\Theta(t)\|$ and $\|\Phi^T(t)\Theta(t)\|$. Here $\Phi^T(t),\Psi^T(t)\in \mathbb{R}^{m\times n}$ and $\Theta(t)\in\mathbb{R}^{n}$. There is a filter ...
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43 views

$L^2$ vs $L^{\infty}$ norm for interpolation

Under what circumstances should I consider utilizing the $L^2$ norm instead of $L^{\infty}$ when interpolating a function based on sample points? Probably related: Does the answer significantly ...
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40 views

Compute the value of the Sobolev norm in $H^{-1/2}$

I am working with finite elements using domain decomposition in 2D and one of the solutions I need to obtain is the co-normal derivative of the solution along a segment that is the intersection of two ...
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1answer
32 views

Derivative of a Radial Basis Function $\nabla_x \frac{1}{1+\|x - y\|^d}$

I want to calculate the following derivative (with respect to x): $\nabla_x \frac{1}{1+\|x - y\|_2^d}$ where, $x,y \in R^D$ and $d$ is some positive integer. $\| \|_2$ is a 2 norm of a vector.
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Question about the norm $\|A \cdot \|_1$

$A \in \mathbb{R}^{n \times m}$, where $n > m$, has independent columns. Consider the vector norm $\| x \|_\Box = \| A x \|_1$. Consider two matrices $M_1$, $M_2$ which preserve this norm, i.e. ...
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24 views

Why is $\ell_{2,1}$ mixed norm non-smooth?

I'm reading about optimization problems involving mixed norms. In particular I'm getting acquainted with the $\ell_{2,1}$ norm. For a matrix $\mathbf{X}$, the $\ell_{\alpha,\beta}$ norm, ...
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27 views

why and when t0 use norm instead of abs and vice versa

What is the difference between the norm and abs of an expression.. as far i understand does ||a - z|| mean norm and |a-z| abs , but what is the difference?
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47 views

Convexity of a certain function connected to the norm

Suppose that we are given two vectors $x,y$ in a normed space $X$. Can we prove in general that the function $$t\mapsto \|x-ty\|$$ is convex? It is certainly the case if the normed space has ...
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26 views

Piecewise absolutely continuous functions: norm implication

I have the following doubt: Consider a sequence of functions $(f_k)$, $k=1,2,\ldots$ which are piecewise absolutely continuous functions $f_k: [a,b] \rightarrow \mathcal{R}^n$ with a finite number of ...
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7 views

Given a p.s.d. matrix $X$ and vectors $a, b$, what is the relationship between $||a-b||_2$ and $a^TXa-b^TXb$?

Given $X \in \mathbb{R}^{N \times N}$, $a,b \in \mathbb{R}^N$ What is the relationship between $||a-b||_2$ and $a^TXa-b^TXb$?
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108 views

Optimization of Frobenius Norm and Nuclear Norm

How to solve the following optimization problem, \begin{equation} \boldsymbol{\hat{x}} = argmin_{\boldsymbol{x}} \frac{1}{2} \| \boldsymbol{x - y} \|_F^2 + \lambda \| \boldsymbol{x} \|_{*} ...
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54 views

Compute the Frobenius norm

I'm trying to compute the Frobenius norm of $L^{-1}$ where $$L^{-1}= I + N + N^2 + ... + N^{n-1}$$ and $N$ is strictly lower triangular. Can anyone suggest some way to do this?
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76 views

Normable topology determined by its restriction to a finite number of factors?

Is it generally true that all norms $\|\cdot\|$ on a finite product of normed spaces $E_1\times\dots\times E_n$ with $\|(0,\dots,0,x,0,\dots,0)\|=\|x\|_i$ where $\|\cdot\|_i$ denotes the norm on $E_i$ ...
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42 views

Saturation of the Babenko–Beckner inequality

The Babenko-Beckner inequality $|| \mathcal F f ||_q \geq C(q,p)||f||_p$ is a well-established theorem. It relates the $q$-Norm of a Fourier transform $\mathcal F f$ of a function $f$ to its ...
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28 views

Norm and Cauchy sequence.

If someone could please show me how to show that if $x_n$ is a Cauchy sequence then $x_n \over ||x_n||$ is a Cauchy sequence as well? Thanks, I hope I've been clear enough.
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56 views

Dual norm of induced norm on matrices

Given a norm on a (finite dimensional) vector space $V$, we can define the induced norm on linear maps from $V \rightarrow V$, by $\|M\|_{I(V)} = \sup\limits_{\|x\|=1}\|Mx\|$. It is also true that ...
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35 views

Norm of Functional on : $c_0 = \{ x \in l^\infty :\ \lim_{k \to \infty}=0 \}$

Take $E = c_0 = \{ x \in l^\infty :\ \lim_{k \to \infty}=0 \}$ and define on $E$ the functional: $$F(x) = \sum_{n=1}^\infty \frac{x_n}{2^n}$$ $\cdot$Show that $F$ is a linear continuos functional on ...
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60 views

Spectral norm upper bound for covariance matrix

Let $\|\cdot\|_2$ be the spectral norm. Let $x_1,\dots,x_n$ be i.i.d. draws from $N(0,S)$. Let $\lambda_1,\dots,\lambda_n$ be some real numbers. Is it true that $$\|\sum_{i=1}^n \lambda_i x_i ...
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59 views

Max - min problem of a quotient of norms

For the $2\times2$ matrix $\begin{bmatrix}4&0\\-3&-5\end{bmatrix}$ Part 1 Find nonzero vectors $u$ and $w$ that maximize and minimize respectively the quotient $||Av|| / ||v||$. Part 2 ...
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How nuclear norm is convex whereas weighted nuclaer norm is not?

In (http://nuit-blanche.blogspot.in/2014/05/wnnm-weighted-nuclear-norm-minimization.html), it is stated that nuclear norm of a matrix $\mathbf{X}$, given as $||\mathbf{X}||_{*}=\sum_{i} ...
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41 views

Greatest norm for the exponential of a polytopic matrix

im new to the forum and I have a question that Im working quite for a while now. I would like to prove that the greatest norm when considering the exponential ...
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77 views

Matrix norm induced by a vector norm.

All matrices are real. $A$ is a matrix of size $n \times k$ with $k < n$ and has independent columns. The function $v(x) = \|Ax\|_1$ is a norm. What is the matrix norm induced by $v$? Is it of ...
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Estimate bounds on Minkowski distance from point to a segment in Lp space

Assumptions Let $L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric), $a,b$ be arbitrary $n$-dimensional points, $c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...
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40 views

Upper bound for integral over boundary in terms of integral over interior

I've encountered quite some papers in which it is simply assumed that $\exists C>0 : \left(\displaystyle{\int\limits_{\Gamma}}((\nabla v)\cdot \hat{\bf{n}})^2d\Gamma\right)^{\dfrac{1}{2}}\leq ...
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74 views

Show that $\lim_{p→∞} ||x||_p = ||x||_∞$

For any $x ∈ \mathbb{R}^n$ and $p ≥ 1$, define $$||x||_p = \left(\sum_{i=1}^n|x_i|^p\right)^{\frac1p},||x||_∞ = \underset{1≤i≤n}{\max}|x_i| $$ Show that $$ \underset{p→∞}{\lim} ||x||_p = ...
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50 views

Generalized Form of Fano's Inequality

The Wikipedia article on Fano's Inequality presents a generalization as follows: Let $\mathbf{F}$ be a class of probability densities with a subclass of $r+1$ densities denoted $f_{\theta^{(i)}}$ ...
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43 views

Find a family of linear inequalities for which the unit ball is a solution set

Find a family of linear inequalities for which the unit ball $A = \{x \in \mathbb R^n \ | \ \|x\|_2 \le 1\}$ is a solution set Would it just be $x_1^2 + ... + x_n^2 \le 1$?
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89 views

upper bound on this matrix norm

What would be the upper bound on the 2-norm (or any norm) of the following matrix product ? Please consider the smallest upper bound. $\|\left(I+BA^T\right)\left(I+AA^T\right)^{-1}\|< ?$ where A ...
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Equality condition for convolution's $L^p$ norm.

Suppose that $1< p< \infty$, $f\in L^1(R)$, and $g\in L^p(R)$ and that $\|f*g\|_p=\|f\|_1\|g\|_p$. Show that then either $f=0$ a.e or $g=0$ a.e I have solved for $g=0$ a.e. if $||f||_1>0$ ...
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51 views

Norm of a mapping which is a linear combination of other mappings

Let $(X, d)$ be a metric space, $(E, || \cdot ||)$ a Banach space, $(AE(X), || \cdot ||)$ - as described below. $AE_0(X) = \{ u : X \rightarrow \mathbb{R} \ : \ u^{-1} (\mathbb{R} \setminus \{0 \}) \ ...
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251 views

Prove that the normed vector space $(S_F,\|\cdot\|_1)$ is not Banach.

$S_F$ is the space of real sequences $\mathbf a=(a_n)_{n=1}^{\infty}$ such that every sequence $\mathbf a\in S_F$ is eventually zero. $\|\cdot\|_1$ is the norm defined as $\|\mathbf ...
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144 views

Show that the following definitions all give norms on $S_F$

$S_F$ is the space of all real sequences $\mathbf a=(a_n)_{n=1}^{\infty}$ such that each sequence $\mathbf a\in S_F$ is eventually zero. Show that the following definitions all give norms on $S_F$, ...
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158 views

Prove that $N_2(\mathbf z)=(\sum_{k=1}^n \overline {z_k}z_k)^{1/2}$ is a norm on $\mathbb C^n$

Here $\mathbf z=\{z_k=x_k+iy_k\}_{k=1}^n, \overline{\mathbf z}=\{x_k-iy_k\}_{k=1}^n \in \mathbb C^n$. So far I have proven the following properties required for $N_2(\mathbf z)$ to be a norm: ...
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51 views

How can I prove maximum norm?

Prove that for a matrix $A \in R^{n \times n}$, we have $$\|A\|_{\infty}=\max_{i=1,...,n}\sum_{j=1}^{n}|a_{ij}|.$$ I know that $$\|A\|_{\infty}=\max\frac{\|Ax\|_{\infty}}{\|x\|_{\infty}}$$ such ...
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37 views

perturbation of trace norm

The definition of trace norm is the summation of singularities of that matrix. I need to calculate the trace norm of matrix with the form $$A = I + r$$ where the ...
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79 views

Trace norm identity (in bra-ket notation)

I came across the following identity in a paper: $$ \|\hspace{0.3em}|v\rangle\langle v| - |w\rangle\langle w|\hspace{0.3em}\|_{tr}=2\sqrt{1-|\langle v|w\rangle |^2}$$ where the norm on the left is ...
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25 views

Prove $ \left \| \left [ G \: \: \: \: I \right ] \right \|^{2}_{\infty }\leqslant \left \| G\right \|^{2}_{\infty } +1 $

Consider the Strickly proper transfer function of G(s) and impulse response G(t), by assuming the input signal u(t) having the bounded $\left \| \right \|_{1}$ , How can one prove the following ...
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98 views

Show linearity of this map

We have the following maps on a complex vector space $V$ $\phi : V \rightarrow \mathbb{C}$ and $g : V^2 \rightarrow \mathbb{C}$ where $\lambda \in \mathbb{C} , x,y,w \in V$. $\phi $ satisfies that ...
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27 views

Finite parameter integral implies finite norm

Need a bit of help with a parameter integral problem. We have, $X$ is a finite measure space with measure $\mu$ and $f:X\rightarrow [0 , \infty)$ is a measurable function. The parameter integral ...