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Questions tagged [norm]

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

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“Norm of norms” is another norm?

Suppose that, for some finite-dimensional real vector space $\Bbb R^n$, that $n_1(v)$, $n_2(v)$, ..., $n_k(v)$ are a set of norms on the space. Given some $v$, then, we can look at the "vector of ...
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28 views

For $g(x) = \int_0^\infty (x+y)^{-1} f(y) \, dy$ show $|g'(x)| \le c_p \frac{1}{|x|^{1+1/p}} \lVert f \rVert_{L^p}$

Q: For $x > 0$ and $f \in L^P(0,\infty), 1 \le p < \infty$, \begin{align*} g(x) &= \int_0^\infty (x+y)^{-1} f(y) \, dy \\ \end{align*} Show that $g(x)$ is continuous and in fact ...
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Infinity norm is actually a norm : triangle inequality

I have to prove the following assertion : Let $V$ be a finit dimentional vector space with dimension $n$ over the field $K$ which is the field of real numbers or complex numbers. Let the map defined ...
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20 views

Is the norm of a vector given by dividing it by the square root of the inner product with itself?

I've known the norm (aka. unit vector in the direction of) a vector (let's call it $v$) to be given by the formula: $v/\sqrt{v \cdot v}$, where $\cdot$ is the dot product. But can this be ...
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1answer
19 views

A convenient redefiniton of the Sobolev norm

I am dealing with the Sobolev space $W^{m,2}[0,1]$, i.e. the space of functions on $[0,1]$ with absolutely continuous $m-1$st derivative and square integrable $m$th derivative. I am using the norm $$|...
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10 views

Deriving the residual error for finite elements.

I am using the following set of notes for adaptive finite elements (https://www.ruhr-uni-bochum.de/num1/files/lectures/AdaptiveFEM.pdf) and am trying to go through the error calculations on page 29&...
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Prove that the sequence $( \frac{1}{n} , \frac{1}{n}) \subseteq \mathbb{R}^2$ converges to $(0, 0)$ with respect to the norm $\|\cdot\|_2$. [on hold]

Prove that the sequence $( \frac{1}{n} , \frac{1}{n}) \subseteq \mathbb{R}^2$ converges to $(0, 0)$ with respect to the norm $\|\cdot\|_2$.
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Why do polynomial norms only consider the coefficients? Rather than the basis as well?

Why do polynomial norms only consider the coefficients? Rather than the basis as well? It seems a bit illogical to say that $\|p\|_1=\sum_{k=0}^n |a_k|$, but e.g. $\|f\|_1=\sum_{j=0}^k \| f^{(j)}\|_{\...
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1answer
55 views

Duality Theorem for Minimum Distance Problems

The minimization of the one-norm can be stated as: $$ \min_{u\in\ell_1} \|u\|_1 \qquad \text{subject to} \qquad Au=b, $$ where $u\in\mathbb{R}^m = [u_1,u_2,...,u_m]^\intercal$ is the sequence that we ...
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Showing isomorphism of two $C^*$ algebras

It seems that quite a standard trick of showing two $C^*$ algebras are as follows: Let $A$ be a $C^*$ algebra $B$ another $C^*$ algebra. $A' \subseteq A$ be a subalgebra that is closed under $*$. (...
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Normal Matrix: Norm vs. Spectral Radius

Define the following matrix norm: $$ ||A||=\frac{\max|\langle Ax,x\rangle|}{||x||_2^2} $$ The spectral radius of $A$ is: $$ \rho(A)=\max_i\{|\lambda_i|:Ax_i=\lambda_ix_i; x_i \neq 0\} $$ Can you ...
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17 views

Relation between residue and closeness of solution for non-linear system $Ax=b$

Let $A(x)\in \mathbb{R}^{n\times n}$ be a matrix depending on $x$, $b\in \mathbb{R}^n$ such that, $$A(x)x=b,$$ i.e. we have a system of non linear equations. Let $u\in \mathbb{R}^n$ be its solution ...
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1answer
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Matrix Norm Inequality Implies Invertibility

Could you prove me some hints to prove the following theorem? If $A$ is a non-singular matrix, and $B$ is a matrix such that: $$ \|B-A\|<\frac{1}{\|A^{-1}\|}, $$ then $B$ is non-singular and that: ...
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Upper bound for induced norm

I would like to obtain a tight upper bound for the following matrix norm: $$ \| I - \frac{x x^T}{\|x\|_2^2} \| $$ where $x$ is a column vector. (Clearly, the second term is a rank-1 normalized ...
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Squared norm of sum larger than sum of squared norms [closed]

Let $G$ be an inner product space. Show that for any $g_1,...,g_n\in G$, there exist scalars $c_1,...,c_n$ s.t. $\forall i, |c_i|=1$ and $$\left\|\sum_{i=1}^n c_ig_i \right\|^2\geq \sum_{i=1}^n\|g_i\|^...
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$U_{p,q}$ is bounded

I am trying to prove $U_{p,q}$ is bounded using the induced norm $|| . ||_2$ from $M_n(\Bbb R)$ (or $M_n(\Bbb C)$ I am not sure). A norm is an application $M_n(\Bbb C) \to \Bbb R^+$, but in the case ...
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1answer
28 views

Proving that $l_\infty$ is complete

I'm learning about Hilbert spaces and operators theory, from some book. I came across the following question - And the books' answer: What I don't understand in the proof - Why can we understand ...
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Simplifying induced matrix norm expressions

Notation Absolute value of $c\in\mathbb{C} = \left|c\right|$. Entrywise absolute value of $A = \text{abs}(A)$. Complex conjugate transpose of $A = A^* = (\overline{A})^T$. $A$'s $i$th singular value $...
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3answers
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Real analysis - norm of bounded limit points

I've been struggling with a real analysis problem for 3 days, and I'd appreciate your help with it. Let $||\cdot||$ be an arbitrary norm in $\mathbb{R}^n$ and let $(x_m)$ be a sequence in $\mathbb{R}^...
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Difficulties with Inner products and polarization identities

I am discussing the general inner product space. Here is what Polarization Identities mean. I denote the inner product by $(x,y)$. I am having a difficult time with the polarization identities. Of ...
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Let $G$ be a “Normed” Abelian Group, is this a topological equivalent Norm over $G$?

Let $G$ be an abelian group. We say $\Vert \cdot \Vert : G \rightarrow \Bbb R_{\ge0}$ is a norm if it satisfies $\Vert x \Vert =0 \Leftrightarrow x=0$ $\Vert -x \Vert = \Vert x \Vert$ $\Vert x+y \...
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Is vectorwise L-1 norm matrix monotone?

I am stuck in a convex optimization problem and need to show this following result for further progress. Suppose, A, B, A-B positive definite. $||vec(A)||_1= \sum_{i,j} |a_{ij}|$, prove or disprove ...
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Euclidean norm of complex exponential

If I have $$\begin{aligned} \hat{\boldsymbol{a}}(z, \omega) &=e^{-i \frac{\omega}{c} z}\left(\zeta^{1 / 2} \hat{\boldsymbol{u}}(z, \omega)+\zeta^{-1 / 2} \hat{\boldsymbol{v}}(z, \omega)\right) \\...
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1answer
23 views

Trace Norm / Nuclear Norm: How to verify?

The nuclear norm is defined by this [from wikipedia]: $$\|A\|_* = \text{trace}(\sqrt{A^*A}) = \sum_{i=i}^{\min\{m,n\}}\sigma_i(A)$$ I get the derivation of this equation. However, I wanted to test ...
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1answer
15 views

$L: \Bbb{R}^n \rightarrow \Bbb{R}^m$ be the linear mapping then $||L|| = $ max$_{1 \leq i \leq m, 1 \leq j \leq n}$ $|a_{ij}|$.

Let $L: \Bbb{R}^n \rightarrow \Bbb{R}^m$ be the linear mapping with the matrix $(a_{ij})$ with $1-$norm on $\Bbb{R}^n$ and sup-norm on $\Bbb{R}^m$. I am trying to show that the corresponding norm on $...
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1answer
69 views

Extending $\|H^{\frac{1}{2}}XK^{\frac{1}{2}}\|\leq\frac{1}{2}\|HX+XK\|$ from matrices to operators

I saw in some literature that many author works in finite dimensional (matrix) is because it can be extended into infinite dimensional (operator). The case is as follows: If the following inequality ...
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Is the space of all infinitesimal numeric sequences complete?

Is the space $c_0$ of all infinitesimal numeric sequences $a = (a_1, a_2,...,a_n,...)$ complete? The norm is $||a|| = sup|a_i|$. As i couldn't prove that every Cauchy sequence has a limit in $c_0$ (...
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1answer
97 views

How to minimize $\| x \mathrm a - \mathrm b \|_1$ without using linear programming?

The following question is a generalization of a question asked earlier today: Given vectors $\mathrm a, \mathrm b \in \mathbb R^n$, can one solve the following minimization problem in $x \in \...
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How to minimize $\| c \mathbf{x} - \mathbf{y}\|_1$ without using linear programming?

Is there a closed form solution to the minimization problem $$\min_{c \in \mathbb{R}}\left\lVert c \mathbf{x} - \mathbf{y}\right\rVert_1$$ where $\mathbf{x} = \begin{bmatrix}0 & 1 & \dots &...
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Variants of The Polarization Identity

A problem in Steele's Cauchy Schwarz Master Class asks the reader to prove these "variants of the polarization identity". Let $\langle \cdot, \cdot \rangle$ be a complex inner product and $\alpha \in ...
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Dual norm of the norm induced by inner product

Given a positive definite matrix A, let $<x, y>_A=x^\top Ay$. This inner product induces a norm $\|x\|_A^2=<x, x>_A = x^\top A x$. My question is, what is the dual norm of $\|\cdot\|_A$? ...
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1answer
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1 and inf norm matrix inequality?

I have succeeded in showing that the inequality $\frac{1}{N} ||\vec{x}||_1 \le ||\vec{x}||_{\infty} \le ||\vec{x}||_1$ and I know that I can extend this to show that a simliar form applies to the $\...
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1answer
37 views

Does anyone know this inequality $\big\|x-y\big\|^2\geq \big\|x\big\|^2+\big\|y\big\|^2-2\big\|x\big\|\big\|y\big\|$

I came across an inequality that I have not seen before. I can not say if this is correct. But two things interest me: Is the inequality correct? Where does the inequality come from? This is the ...
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Exist a constant that only depends of a power and dimension that bounds the euclidean norm.

I have this problem but I don't know how to solve it. Considering the Euclidean norm |x|, in $\mathbb{R}^n$. If $a>0$, prove there exist $c>0$ depending only of $n$ and $a$ such that: $$c^{-1}(|...
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1answer
47 views

L1 norm minimization over a matrix for a linear system

Let $\mathbf{A} \in \mathbb{R}^{m \times n}$, where $m<<n$ and $\mathbf{b} \in \mathbb{R}^{m}$. The rank of $\mathbf{A}$ is $m$ and both $\mathbf{A}$ and $\mathbf{b}$ are known. Consider the ...
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1answer
40 views

$||\textbf{A}\textbf{B}||_2=||\textbf{B}\textbf{A}||_2$?

Let $\textbf{A}$ and $\textbf{B}$ be $n\times n$ matrices. Is it true that $||\textbf{A}\textbf{B}||_2=||\textbf{B}\textbf{A}||_2$? I tried to prove this by the following argument: $$\text{det}(\...
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Separating vectors and minimization of norm

I am solving the following exercise from the book Understanding Machine Learning (ex 14.3), the problem is I am not very strong in geometry: Let $S=((x_1,y_1),\ldots,(x_m,y_m)) :\, (x_i,y_i) \in \...
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1answer
29 views

Sup Norm $|| x||_∞$

If $X,Y \in \mathbb R^P \ with \ p\in \mathbb N $ Is true that $|X \cdot Y| \leq || x||_∞ || y||_∞ ? $ I know that $|| x||_∞= Sup( |x_1|,|x_2|,...,|x_n|) $ I don't know how to start this ...
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26 views

Inner Product norm squared

In a book for quantum information, I found the following expression: $$\sum_{x_0\in\{0,1\}^n}4\big\|\langle x_0|\phi^k\rangle|x_0\rangle\big\|^2=1$$ If so, would not it be 4 as a result? Now, I ...
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Proof of norm of a solution of a differential equation is less than the norm of equation itself.

Consider the problem $$-eu''+xu'+u=f$$ $x$ is defined on the interval $I=[0,L]$. $u(0)=u'(L)=0$ where $e > 0$ is a constant. Prove that the solution satisfies $||eu''||≤ ||f||$ where norm is the $...
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1answer
28 views

Inner product is bounded above by product of $\infty$-norm and $1$-norm

Show that if $u,v \in \mathbb R^m$, then $|u\cdot v|\leq \|u\|_\infty\|v\|_1$. By Cauchy-Schwarz, $$|u\cdot v|\leq \|u\|_2\|v\|_2$$ Note also that $\|u\|_1\leq \sqrt{m}\|u\|_2 \leq m\|u\|_\infty$, ...
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1answer
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The angle between the $x$-axis and the vector that reachs the $2$-norm of a matrix

Suppose we have a matrix $A\in \mathbb{R}^{2\times 2}$. We have the $2$-norm defined as $\left \|A\right \|_2=\max_{\left \|x\right \|_2=1}\left \|Ax\right \|_2$. We know that in order to find a unit ...
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1answer
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Subordinate matrix norm inequality in research paper where authors replace $||A||_{op}^2$ with $||A^TA||_{op}$

I'm perusing this paper. In page 8, I came across this: My question is about equation 45. Also in this question I don't care about neither $u$ nor $M$, I mentioned them just to give some context. In ...
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2answers
40 views

Triangle inequality squared?

I am in the process of understanding a proof. There, for example, the following is indicated: $$\big\||a\rangle+|b\rangle\big\|^2\leq\big\||a\rangle\big\|^2+2\big\||a\rangle\big\|\big\||b\rangle\big\|...
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69 views

Computing the derivative of $\|Ax\|_2$

Compute the following derivative (in matrix form) $$\frac{\partial\, \|Ax\|_2}{\partial x}$$ where $A$ is an arbitrary matrix and $x$ is a vector. I think somebody said that the result is $2A^TAx$, ...
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2answers
27 views

Spectral norm and inner product

We know that for a general $N\times n$ matrix $A$, its spectral norm is defined as $$\|A\|=\sup_{x\in S^{n-1}} \|Ax\|_{2},$$ where $x$ is from the unit sphere. My question is that why when $A$ ...
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0answers
23 views

Confusion concerning the Euclidean Norm and unitary invariance on $\mathbb C^n$

For a unitarily invariant norm on $\mathbb C^n$, how do I show that $||x||=||x||_2||e_1||$? I can show that $||e_1||=1$ for the Euclidean norm by definition, and is therefore unitarily invariant, does ...
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1answer
16 views

Finding norm of a bounded linear functional

Let $X=\{x\in C[0,1]: x(0)=0\}$ with sup norm and let $f:X\to \mathbb{K}$ be defined by $$f(x)=\int\limits_0^1 x(t)dt \text{ for all }x\in X.$$ I want to show that $\|f\|=1$. It is easy to show that $\...
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1answer
51 views

Prove $\langle u,v \rangle = \frac{1}{4} \sum^4_{k=1}i^k \left \Vert u+i^kv \right \Vert$

For an assignment in one of my math classes I have this problem. Here is where I have gotten so far. This is in V, an inner product space over $\mathbb{C}$ $$\frac{1}{4} \sum^4_{k=1}i^k \left \Vert u+...
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1answer
46 views

Solve: $\|u+v\| \le \|u\| + \|v\|$ with $\|x\| = \left( \sqrt{|x_1|} + \sqrt{|x_2|} \right)^2$

I was given the following task: Check if $x\rightarrow \left(\sqrt{|x_1|} + \sqrt{|x_2|}\right)^2$ is a norm on $\mathbb{R}^2$. I've already shown that $$\|x\| \ge 0\qquad \|x\| = 0 \...