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

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Does $f_n$ converge to $f$ uniformly?

Consider $f_n: [0,1]\longrightarrow\mathbb{R}$ is given by $$f_n(x) = \begin{cases} \sqrt{n}, & \quad 0<x<\frac{1}{n} \\ 1, & \quad \text{otherwise.} \end{cases}$$ 1.) ...
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25 views

The norm of a vector with gaussian noise

Say I have a vector of length n, $v \in R^n$ where $0<=v(i)<=1$ for each i, Now, I add noise: let $n$~$Normal(0,\sigma)$ And to each i I add noise v(i)+n , such that the noises are independent ...
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25 views

Why this relation in Hilbert space (with inner product $< >$) holds?

$c_k$, $f_k$ are sequences in Hilbert space, $g$ is a function. Why this relation below holds? How you derive it? $\sum_k|c_k\left \langle f_k,g \right \rangle|\leq(\sum_k|c_k|^2)^{1/2}(\sum_k|\left ...
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33 views

Conditions on $\alpha_n$ for $n_\alpha(x)=\sum^\infty_{n=1} \alpha_n\vert x_n\vert $ to be a norm on $l_p$

When $n_\alpha(x)=\sum^\infty_{n=1} \alpha_n\vert x_n\vert $ is a norm in $\mathcal{l}_p=\lbrace (x_k)^\infty_1 : \sum\vert x_k\vert ^p \lt\infty\rbrace $ and $\alpha\in\omega$. and $\omega$:space of ...
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32 views

Relation between the weighted matrix norm and the weights

For a nonsingular matrix $W \in \mathbb{C}^{m\times{}m}$, the weighted vector norm is defined as $||\overrightarrow{x}||_W = ||W\overrightarrow{x}||$. Let $||A||$ denote the induced matrix norm by the ...
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25 views

Analytical solution to fitting two functions

I have two oscillatory functions $f(x)$ and $(k x)^2 g(x)$ where $f$ and $g$ are known and it is also known that the two functions are approximately similar. How can I analytically find the best ...
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34 views

Pareto distribution and matrix

I am wondering if there are any bounds are known on the eigenvalues of random matrix whose entries are with Pareto distribution? Thank you.
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51 views

$l^p$ is super subset of $ l^q$ if and only if?

$$ l^p \subseteq l^q. $$ if and only if ? $l^p$ is a subset of $l^q$ when is it possible ? !
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31 views

Matrix norm of product equal implies equality in norms of factors

Given a matrix $A$, if $$\|Av\|_1=\|Aw\|_1$$ for given vectors $v$ and $w$, then does $\|v\|_1=\|w\|_1$? Here $\|\,\cdot\,\|_1$ denotes the $L^1$ norm.
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How to prove that $\ell_p$ norm is smaller than $\ell_q$ norm if $p > q$?

Let $x \in \mathbb{R}^n$ be an $n$-dimensional vector. It is apparently well-known that, \begin{equation*} \Vert x \Vert_p \le \Vert x \Vert_q \end{equation*} for any $p > q \ge 0$. (cf. ...
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19 views

Estimation of $\alpha_{0}$ using norm of $y_{0}$ and $y_{1}$

If $$y_{0}=\frac{1}{5}x^{5}+\frac{1}{2}x^{4}+\frac{2}{3}x^{3}+x^{2}+2x+1$$ and ...
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16 views

The spectral norm of $ \begin{bmatrix} F_{k+1} x & F_{k}\\F_{k} x^2 & F_{k-1} x\end{bmatrix} $

In mathematics, the matrix induced norm is a natural extension of the notion of a vector norm to matrices. In the special case of $ p = 2$ (the Euclidean norm) and $m = n$ (square matrices), the ...
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21 views

What happens to $l_1$ if i change coordinate system.

Let $x =(x_1,\ldots,x_n) \in \mathbb{R}^n$ and also $x=\sum_{i=1}^m t_i u_i,$ where $t_i \in \mathbb{R}$ and $u_i \in \mathbb{R}^n.$ Is it true that $||x||_1 \geq \sum_{i=1}^m |t_i|$ ?
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46 views

is there any infinity norm bound to simplify this

I have a problem of the form $$\sup_{x\in\Bbb{C}^n}\left\{\frac{\|Ax\|_\infty}{\|Bx\|_\infty}\right\}$$ where $A$, $B$ are matrices with different number of rows and $x$ is an $n$ dimensional vector. ...
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83 views

comparison of 3 topologies on C[0,1]

I have a ring of continuous functions from $[0,1]$ to $\Bbb R$. And two norms $C[0,1]\to\Bbb R$. One is supremum of $|f(x)|,$ the other the value of $\int_0^1|f(x)|$. Then I get a Cartesian product of ...
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52 views

about vectors norm

in the following article http://blanche.polytechnique.fr/~mallat/papiers/MallatPursuit93.pdf page 3 he say: $$y= \langle y , a_{k_0} \rangle a_{k_0} + R $$ with $a_{k_0}\in D$ with $\forall ...
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34 views

Injective norm on tensor algebra of a finite-dimensional Banach space

Suppose that $V$ is a finite n-dimensional Banach space, and suppose that $T(V)$ is the tensor algebra on it. Furthermore, suppose that $T^{(n)}(V)$ is the "abridged" tensor algebra obtained by taking ...
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40 views

How to detect reflexivity of the closure

Consider the space of continuous bounded functions on a bounded interval. Its closure for the Lebesgue $L_p$ norm is reflexive when $1 < p < \infty$, but it is not reflexive for $p = 1$. How ...
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76 views

A matrix in $SL(2)$ has it's supremum norm and infimum fulfilled by orthogonal vectors

I am having trouble proving the next statement: If $B\in SL(2)$ and $||B||\neq 1$, for $||B||:= \underset{x\neq 0}{\sup}\big\{\frac{||B(x)||}{||x||} \big\} $, where $||\cdot||$ is the euclidian norm, ...
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39 views

Why all norms define the same relative interior?(Convex Optimization, Stephan Boyd)

When I was reading 'Convex Optimization', Stephan Boyd, I was stopped by Example 2.2. Before Example 2.2 is started, following definition is coming. If the affine dimension of a set ...
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157 views

Prove that the square sum of eigenvalues is no more than the Frobenius norm for a square complex matrix

Prove: $$ \sum_{r=1}^{n} |\lambda_r|^2 \le \sum_{i,j=1}^{n} |a_{ij}|^2 $$ the equality holds if and only if $\boldsymbol{A^H A=AA^H} $ for a square complex matrix $ ...
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32 views

Quadratic Time-Frequency Representation with L2 norm

I have been reading the book Multidimensional Particle Swarm Optimization for Machine Learning and Pattern Recognition. They use different norm for different problems in their automatic ECG detection ...
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78 views

Contraction mapping and $L^{2}(\Bbb R \times \Bbb R)$ spaces in inequality

I found this: \begin{equation} \lVert Wx \rVert^{2}_{L^{2}( \mathbb{R} \times \mathbb{R} )} \leq \lVert x \rVert^{2}_{L^{2}(\mathbb{R} )}, \end{equation} which I think ...
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51 views

Understanding the matrix pertubation.

I would like to ask a question about perturbation in matrix, as I understand from this tutorial. Main task of perturbation is to determine what amount of minimum change is necessary in given ...
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27 views

Why is the tensor product of admissible homomorphisms of seminormed A-modules admissible?

Let $A$ be a normed ring, and let $M$ and $N$ be seminormed $A$-modules. If $\phi:M'\rightarrow M$ and $\psi:N'\rightarrow N$ are admissible homomorphisms of seminormed $A$-modules, then I want to ...
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146 views

Gradient of a norm with a linear operator

In mathematical image processing many algorithms are stated as an optimization problem, where we have an observation $f$ and want recover an image $u$ that minimizes a objective function. Further, to ...
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Normalized cross-correlation in detail

I'm trying to implement a normalized cross-correlation algorithm but I don't get what in fact is this measure. What confuses is the wikipedia definition: $\frac{1}{n} \sum \frac{(f(x,y)- ...
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126 views

L2 Norm of Gaussian Integer

I'm writing a Java program that deals with Gaussian Integers. In my program, I have to compute the L2 norm of the GI (Gaussian Integer) and return it as a float. I've looked around but I cannot seem ...
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55 views

What is the norm of v + w

So the norm of v is defined as ||v|| = sqrt(v_1^2+...+v_n^2) I would guess that the norm of v+w is defined as ||v+w|| = sqrt((v_1+w_1)^2+...+(v_n+w_n)^2) Is this correct? My textbook asks me to ...
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51 views

Frobenius Norm Unitary Operators

For something I'm working on, I have a matrix $A$ with other matrices $U$ and $V$ which are unitary ($U^*U = I$ and $V^*V = I$), and I'm trying to show that, for the Frobenius norm, $\|A\| =\|UA\| ...
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20 views

Following positive semi-definitness from matrix norm

How can I follow the following? $$||A||_2 \le \sigma > 0$$ $$\Leftrightarrow A -\sigma I \mbox{ is positive semi-definit}$$ I always get it the other way around, i.e. that $\sigma I - A$ is ...
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30 views

Methods of computing the derivative of vector norms

I am very new to norms. Except the basic definitions and properties of the norm, I don't know too much about it. Now, I am very interested in computing the derivative of the norms. So, I am wondering ...
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36 views

If a matrix of the form I + B is singular, then ||B|| ≥ 1 for every subordinate norm.

I need some guidance showing that: If a matrix of the form I+B is singular, where I is the identity matrix, then for any subordinate norm $\|\cdot\|$, $\|B\|\geq1$.
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Norm of the maximum

Consider the norm $||f||= max_{x\in[a,b]} |f(x)|$ defined in the bectorial space $C[a,b]$ I have to what is the meaning (/interpretation) in $R$ of {$||f_n-f||$}$\to$ 0 Could you help me?
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33 views

Square of 2-norm

This might be silly but I am stuck with the following problem: $ || Y - Z_i/x||^2_2 $ = 2t How would I solve to get $x $ from this equation?
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41 views

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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145 views

Finding a Unit Vector v for a Matrix A such that the 2-norm of AV is equal to the 2-norm of A

I have been working on the following problem: Let A be the following 2x2 matrix: A = [1 1; 0 1] (MATLAB notation) Find the 2-norm of A and a unit vector v such that the 2-norm of Av = the 2-norm of ...
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1answer
67 views

bounding operator norm by quadratic forms on orthonormal basis

Let $\{u_i\}_{i=1}^n$ be a set of orthonormal basis and $M$ a symmetric matrix. Suppose \begin{equation} \left|\langle u_iu_i^T,M\rangle\right|\leq \tau\ \ \forall i=1,\dots,n. \end{equation} Can I ...
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How to prove $||A||_2\leq ||A||_F \leq \sqrt{n}||A||_2$ [duplicate]

$A$ is a square matrix with dimension $n$ and $||A||_F$ is Frobenius norm.
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Infinite number of induced norms

While proving that a norm had to come necessarily from a scalar product I have started to wonder about the concept and uniqueness of induced norm. My teacher hasn't clarified to me this doubt, saying ...
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51 views

Maximum coordinate of a linear transformation of a vector

Given a vector $x \in R^n$ (variable) and a constant matrix $M \in \{0, 1\}^{m \times n}$ (known). $M$ is a binary matrix, meaning that its entries are either $0$ or $1$. I need to obtain an ...
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74 views

Difference: normed space and normed linear space.

I'm currently reading up on Banach spaces. My book "Introduction to Banach spaces and their Geometry" by Beauzamy mentions "normed spaces" in some places, and "normed linear spaces" in other. I really ...
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31 views

Norm of difference of two squares of matrices

Let $x,y$ be square matrices and $c$ be any scalar. Is it true that $ \Vert x^2 \Vert - c^2 \Vert y^2 \Vert = \Vert x - cy \Vert ^2$? If this is true then I'm done with the proof of a theorem on ...
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41 views

Norm of a mapping

$$C[0,1]=\{f:[0,1]\rightarrow R | \text{$f$ is continuous function}\}$$ $$\|f\|=\max \{|f(x)|:x\in [0,1]\}$$ $$A:(C[0,1],\|\|)\rightarrow(C[0,1],\|\|)$$ $$A(f)(x)=(x^4-x^2)f(x)$$ I have to ...
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87 views

Proving nonsingularity of a sum of matrices

I'm trying to solve this study question but I'm not sure how to proceed. The question is as follows. If \begin{equation}\frac{||B||_2}{||A||_2}<\frac{1}{\kappa_2(A)}\end{equation} with ...
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69 views

Extrema of a vector norm under two inner-product constraints.

If $\langle\vec{A},\vec{V}\rangle=1\; ,\; \langle\vec{B},\vec{V}\rangle=c$, then: \begin{align} max\left \| \vec{V} \right \|_{1}=?\;\;\;min\left \| \vec{V} \right \|_{1}=? \end{align} Consider the ...
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319 views

Norm of a vector-valued function?

When studying commutator estimates, I have encountered the following problem. Consider $f\in C^1(\mathbb{R}^d,\mathbb{R})$ with $\nabla f\in L^p$. So $\nabla f(x)\in\mathbb{R}^d$. My question is ...
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716 views

How to describe minimization of L1 norm error using linear programming?

Given a set of $n$ pair points $(x_1, y_1), ..., (x_n, y_n)$ in the plane, I need to find a line $ax + by = c$ that fits the points of the L1 norm error points as closely as possible. I need a linear ...
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1answer
53 views

SVM - Min square norm

All Support Vector Machine litterature mentions that optimal hyperplane is found as: max 1/∥x∥ (st. constraints) which translates directly to: min ∥x∥ or equivalently min $ ∥x∥^2 $. Here ...
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43 views

Natural invariant norm on the space of polynomials

What is the " Natural" invariant norm on the space of polynomials in a complex variable $z$? And can anyone give me an idea as to how it is deduced?