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

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$L^p$ and $L^1$ norms

As $L^p$ norm of $f:\mathbb R\to \mathbb R^{n}$ is defined as $$ \|f(t)\|_p:=\left(\int_S\|f(t)\|^pdt\right)^{1/p} $$ I have two questions: What kind of norms for the function $f(t)$ in the ...
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Is this a property of Normed Vector Spaces?

Let $X$ a normed vector space on $(\mathbb{K}, + , . )$. Is the following assertion true? Any $x$ of $X$ can be written as $x = \alpha a$ , $\alpha \in \mathbb{K}$ , $a \in X$ with $||a||_X=1$ ...
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49 views

Are the infinity norm and the the induced norm equivalent in $l^2$?

Consider the function $||\cdot||_\infty:l^2\to[0,\infty)$ given by $$||(a_n)||_\infty = \sup\{|a_n|:n\ge 1\}.$$ So I want to know if this norm is equivalent to: $$||(a_n)||=\sqrt{\sum |a_n|^2}$$ ...
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59 views

Open Ball and Lipschitz Equivalence equivalence

I am trying to show that two norms $\|\cdot\|$ and $\|\cdot\|^\prime$ are Lipschitz equivalent if and only if there exist numbers $r,R >0$ such that $B_r \subseteq B_1^\prime \subseteq B_R$ where ...
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Lower bound on multiplicative norm of Banach algebra.

Let $A$ be a Banach algebra over $\mathbb{C}$ and $N: A \to \mathbb{R}_{\ge 0}$ the corresponding multiplicative norm. For $a \in A$, do we have$$\limsup_{n \to \infty} (N(a^n))^{1\over{n}} \le N(a)?$$...
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31 views

Geometry of a Cauchy sequence in a normed space

A sequence in a normed space $X$ is called a Cauchy sequence if and only if for every $\epsilon > 0$ there exists an integer $N\in \Bbb N$, such that $\|x_n-x_m\|\lt \epsilon$ for all $n>m>N$....
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Deducing equivalence between norms from simple condition

Let $||\cdot||,\;||\cdot||'$ be norms on $V$. Suppose for some $a,b>0$ we have: $$||x||<a\Rightarrow ||x||'<1\Rightarrow ||x||<b$$ Show that $||\cdot||,\;||\cdot||'$ are Lipschitz ...
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17 views

Understanding the Quotient norm, constructed example

I just read the definition of the quotient norm, but it had no examples, so I constructed my own. Is this correct? Let $(X=\Bbb R^3,\|\cdot\|)$ with Euclidean norm and $Y\subset X$, $\quad Y=\{(a,0,0)...
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19 views

Spectral Radius of A+B

I am solving a physical problem numerically which gives three real, symmetric and positive semi-definite matrices: $A$, $A_1$, and $A_2$; where $A=A_1+A_2$. I know that the following identities ...
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447 views

How to calculate expected value of matrix norms of $A^TA$?

Let $A$ be a random $m$ by $n$ rectangular sign matrix, chosen uniformly at random, with $m < n$. To be clear, $A$ is a matrix whose entries are chosen from $\{1,-1\}$. Let $B = A^T A$. We know,...
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173 views

Uncountably many norms such that no two are Lipschitz equivalent

I am struggling with the following question: Is it possible to find uncountably many norms on $C[0,1]$ such that no two are Lipschitz equivalent? I had thought about trying to define norms for each ...
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20 views

well-defined and continuity of an operator $A:l^{\infty}(\mathbb{Z})\to l^{\infty}(\mathbb{Z})$

I'm stuck to prove that $A:l^{\infty}(\mathbb{Z})\to l^{\infty}(\mathbb{Z}),\; u\mapsto Au$ with $(Au)_j=\sum_{k\in\mathbb{Z}}\rho_{j-k}u_k$ with $\rho\in l^1(\mathbb{Z})$ is well-defined and ...
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80 views

Frechet Derivatives of a nonlinear integral operator

The nonlinear integral operator $P:C[0,1]\to C[0,1]$ is defined as follow: $$P(f)(x)=1+kxf(x)\int_0^1\frac{f(s)}{x+s}ds$$ In order to obtain the Frechet derivative of the operator, I start with: $$...
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Definition of renorming of a space

The following is Proposition $4.5$ from Kalton's paper. Let $X$ and $Y$ be Banach spaces such that there exists a Lipschitz embedding $L:X \rightarrow Y$ such that $L(0)=0$ and $\overline{span(...
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92 views

L2 norm of 2 Normally distributed variables

Given: $Z=\sqrt{X^2+Y^2}, X\sim N(\mu_x,\sigma_x^2), Y\sim N(\mu_y,\sigma_y^2)$ What is the expected value of $Z$? I'm specifically looking for the case where the $\mu_i$ are non-zero and $\sigma_i$ ...
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54 views

Dot product and a norm

Let $\langle\cdot,\cdot \rangle$ be a dot product on $\mathbb{R}^{2}$. We define a norm $\|x\|=\sqrt{\langle x,x \rangle}$. We know that: $$ \sup_{x \in \mathbb{R}^2}{\frac{\| x\|_2}{\|x\|}}=3 ,\...
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15 views

Two-norm function norm of interpolation

I am attempting to calculate the two-norm of this function $f_k(x)$. $$ \|f_k\|_2 = \left( \int_0^1 f_k(x)^{2} dx \right)^{1/2} $$ The answer is listed as $$ \sqrt{2/3} $$ However, I'm not getting ...
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29 views

$\|A\begin{bmatrix} m\\n \end{bmatrix}\| = \|B\begin{bmatrix} m\\n \end{bmatrix}\| $, what can be said about A and B?

Given two 2x2 matrices A and B such that for some real m and n: $$\|A\begin{bmatrix} m\\n \end{bmatrix}\| = \|B\begin{bmatrix} m\\n \end{bmatrix}\| $$ The norm is euclidean, and $A\ne B$ and $m\ne 0$ ...
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79 views

Derive the dual function $g(\lambda, \nu)$ for the least-norm problem

I am trying to find the dual function $g(\lambda, \nu)$ to this problem $$\min\limits_{Ax = b} \|x\|$$ Step 1. Form the Lagrangian $$L(x, \lambda, \nu) = \|x\| + \nu^T(Ax-b) = \|x\| + \nu^TAx - \...
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24 views

Am I showing the positive-definiteness of a symmetric matrix A correctly?

A is a 4x4 square, symmetric matrix. First I computed its Cholesky Decomposition $LL^t$ since the first part of the question asked for it. Then it asked to show A was positive-definite. I am a ...
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27 views

Orthogonal matrices and different norms

I'm trying to understand the relation between orthogonal matrices and different norms through a few practice problems, but as they don't have a solution guide I've gotten particularly stuck on one. ...
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$|Ax|≤\|A\||x|\space\forall x\in\mathbb{R}^n$ (Rudin's Principles)

In Rudin's Principles of Mathematical analysis p. 208 $\|A\|$ is defined as the $\sup$ of all numbers $|Ax|$, where $x$ ranges over all vectors in $\mathbb{R}^n$ with $|x|≤1$. Then he claims $$|Ax|≤\...
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64 views

Find norm of operator $L(x,y)=(x+3y,y-x)$

I'm trying to tackle the following question, but with no success... Let $L: \ \mathbb{R^2}\to\mathbb{R^2}$ be an operator such that $L(x,y)=(x+3y,y-x)$. Find $\|L\|$. So, I know that I need ...
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Why is $\max_{1≤i≤m,1≤j≤n}|a_{i,j}|≤||A||_{op}$

Why is $$\max_{1≤i≤m,1≤j≤n}|a_{i,j}|≤||A||_{op}$$ where $$||A||_{op}=\sup\{||Ax||\space |\space x\in\mathbb{R^n}, ||x||=1\}\space\space\text{(operator norm)}$$ ?
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32 views

arg min-invariance for norm of vectorfield under linear transformation

Given a vectorfield $\vec{F}(\vec{c}) \in \mathbb{R}^n$ which is a function of some parameters $\vec{c}$, what constraints must you have on a matrix such that when you act on the vectorfield the $\arg\...
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How can I prove this proposition of linear algebra?

Good afternoon! I have to show this proposition: 1) Let $A \in \mathbb R^{n\times n}$ a non singular matrix and $PA=LU$, $P$ permutation matrix, $L$ a lower triangular matrix with $1$ on its ...
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“Decay” of $L^2$-norm of the solution of heat equation with mixed boundary conditions

I'm considering the heat equation $$u_t = u_{xx}$$ on the interval $[0, \pi]$ with mixed boundary conditions $$u(0,t)=0 \quad \text{and} \quad u_x (\pi,t)=0, $$and smooth initial data $u(x,t)=u_0 (x)$ ...
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Min (or Max) $L_1$ norm of Hermitian matrices with given eigenvalues

Suppose the real numbers $(\lambda_1, \ldots, \lambda_n)$ are given. All Hermitian matrices that have eigenvalues $\lambda_i$, $i=1,\ldots,n$, can be represented in the following way: $H = \sum_{i=...
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The inclusions between unit balls in $\ell^p$ spaces

I need to show that $d_∞(x, y) ≤ d_2(x, y) ≤ d_1(x, y) ≤ nd_∞(x, y)$ where $d_1=|x_1-0|+|y_1-0|$ and I'm setting $|x_1-0|+|y_1-0|<0$. Illustrating the $B(0,1)$ balls (centered at 0 with radius 1) ...
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What does a norm which is symmetric “around” an index subset look like?

I am looking at norms $\lVert\cdot\rVert$ on $\mathbb{R}^{n}$ that have the following symmetry property $$\forall \beta \in \mathbb{R}^{n} \text{: }, \lVert\beta \rVert=\lVert \beta_{J}+\beta_{J^{c}}\...
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Householder QR problem

Can somebody give me a hint or help me to solve this problem. Let V be a p×q matrix with orthonormal columns (p > q), and $M = I−2VV^T$ , with I being the p × p identity matrix. The matrix M can ...
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Norm of a full rank matrix and its transpose

If A is p × q with rank q, prove that $||\ A(A^TA)^{−1}A^T ||\ _2 = 1$. What does $A(A^T A)^{−1}A^T$ represent?
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Norm of orthogonal matrices

Can someone help me with this problem. I have no idea how to solve it!! If A is a p×q matrix, U is a p×p orthogonal matrix, and Z is a q×q orthogonal matrix, prove that $||A||_2=||UAZ||_2$
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L1-norm minimisation

I am working on this tutorial question. The question asks me to write a Matlab code to implement the method. I was stuck in how to formulate a code for the proximal operator as well as the gradient ...
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25 views

Norm of a matrix evaluation

In the following A denotes a p×p matrix, and x a p×1 vector. (a) Is $f_1(x) = ||Ax||^2$ a norm on the space of p×1 vectors? What are the conditions (if any) that the matrix A needs to satisfy for ...
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If A is invertible and $||B-A|| < ||A^{-1}||^{-1}$ prove $B$ is invertible.

Just having really hard time trying to proof : If $A$ is invertible and $||B-A|| < ||A^{-1}||^{-1}$ prove $B$ is invertible. It is related to Neumann Series but i don't understand how to proof ...
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252 views

Why is the norm convex?

Why is the norm a convex function?
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Condition number of input y and output x

Take y=area of a rectangle with corner vertices [0, 0] and $[x_1,x_2]$, where input is $[x_1,x_2]$, output is y and input norm is $||_2$, output is absolute value. How to calculate the condition ...
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How to design a strict constraint term (never violate this constraint) for matrix decomposition

It is a low-rank sparse decomposition problem: $ D= ML + S + \epsilon$, that we know the matrix D can be decompose into 2 part that one part $L$ is low rank, the other part $S$ is sparse, $\epsilon$ ...
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Confusion about norms

I was doodling at the kitchen table this morning and I seem to have thoroughly confused myself about convex functions and norms. A norm $||x||$ is a convex function, via the triangle inequality. ...
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dual pairs in $\mathbb R^n$ with non-standard norms $l_1$ and $l_{\infty}$

Suppose $X=Y=\mathbb R^n$. Usually, to apply a separating hyperplane theorem in $X$ (the primal space), we associate both $X$ and $Y$ with the standard Euclidean norm. My question is that could I ...
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Bounding the spectral norm of the inverse of a matrix sum (useful bounds, tightest not necessarily needed)

Let $A$ and $B$ be $n\times n$ matrices. Suppose $A$ is invertible and that $\|A\|_2,\|B\|_2,$ and $\|A^{-1}\|_2$ are known. Are there any useful bounds (upper and/or lower) $||(A+B)^{-1}||_2$ in ...
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34 views

Can we prove that $\lim \|X^T A X\| / \|X\| = 0$ as $\|X\|\to 0$ for all norms?

Suppose $X$ and $A$ are two matrices of compatible dimensions. Is it possible to prove that $$ \lim_{\|X\|\to 0} \frac{\|X^T A X\|}{\|X\|} = 0, $$ where: A is squared, X may not be squared. The ...
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Law of cosines is equivalent to a dot product identity

In this lecture, starting at around 18:00, it's shown the dot product identity $$a^T b=||a||_2 ||b||_2 \cos \theta$$ where $\theta$ is the angle formed b/w $a,b\in \mathbb{R}^3$, is the same as the ...
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How to check whether the following functions are norms on the corresponding spaces?

$\|f\|$:=sup$_{x\in[0,1]}$$\quad \frac{\vert f(x)-f(0)\vert}{x}$ on the space C[0,1] and the same function but on the subspace V of $C^1$[0,1] of functions which equal zero at x=$\frac{1}{2}$. How ...
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52 views

Prove by Induction that Norms in a Finite Dimensional Space are Equivalent?

I would have thought that this is a good candidate for an inductive proof, but I have searched for one and failed. Is there such a proof, and if not why not ? Here's how far I got. It's easy to ...
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46 views

$p$-norms inequalities [closed]

I got this math homework and I can't do it. I have to prove the inequalities for p-norms in $\mathbb R^n$: $\|x\|_1\ge\|x\|_2\ge\|x\|_\infty$ $\|x\|_1\le\sqrt{n}\|x\|_2$ (I have already proven this ...
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80 views

Compactness of the unit sphere in finite-dimensional normed vector space

In order to prove that norms defined on any finite-dimensional real (or complex) vector space $E$ are equivalents, I need to proof the compactness of the unit sphere $S_{\infty}=\{x\in E\,\vert\,||x||...
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Completeness of the space $C_{2\pi} \left ( \mathbb{R} \right )$ [duplicate]

Denote $C_{2\pi} \left ( \mathbb{R} \right )$ is the set of complex-value, continuous on $\mathbb{R}$, periodic functions with period $2\pi$. Fix $p \in \left [1, \infty \right )$. For each $x \in C_{...
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81 views

soft thresholding derivation with two terms

I am trying to minimize the following function $$argmin_x||x-y||^2_2 + ||x-z||_2^2 + \lambda || \frac{x- w}{c}||_1$$ I have been doing it for while but i am not sure how to do it. I have already been ...