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 this question make any sense - eigenvalues and norms

Im having difficulties understanding this question: show that if $b$ is an eigenvector of an invertible matrix $A$ with an eigenvalue $\lambda_1$ and $\delta b$ is an eigenvector of $A$ with an ...
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81 views

proving uniform convergence for a uniformly continuous function

I've been stuck on this question (2a) for about a day now! Question and havn't really gotten anywhere, I think that the series is that integral of a step function that approximates f ie: $\Gamma_n ...
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385 views

Triangle inequality frobenius norm

I'm trying to show that the frobenius norm is a norm. however it appears as if triangle inequality isnt met. $$||A+B||_F = \sqrt{\sum_{i,j=1}^n |a_{ij}+b_{ij}|^2} \leq \sqrt{\sum_{i,j=1}^n ...
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130 views

Norm inequality with wedge product

Anyone could help me to prove this following inequality? $\displaystyle\frac{||(u+v)\wedge w||}{||u+v||}\le \frac{||u\wedge w||}{||u||} +\frac{||v\wedge w||}{||v||} $ where $u\wedge v$ is the wedge ...
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48 views

Uniformly Bounded in a Schauder Norm

Suppose $\{h_{k}\}$ is a sequence of elements in $C^{2,\alpha}$. I want to show that $\{h_{k}\}$ is uniformly bounded in $C^{2,\alpha}$. Now, if I'm not mistaken, then $$||h_{k}||_{\displaystyle ...
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77 views

Topological equivalence of any norm on $\mathbb C^n$

In University I have been told that every norm on $\mathbb C^n$, for any $n\in\mathbb{N}$, is equivalent to every other such norm. I have a proof for this on any vector space on $\mathbb R$. Trouble ...
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62 views

What is the usual norm of $\mathbb{H_0^1}$?

What is the usual norm of product of sobolev spaces $\mathbb{H_0^1}=H_0^1 \times H_0^1=W^{1,2}\times W^{1,2}$? In my work i need to prove that the norm endowed by the inner product ...
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38 views

Equivalence of norm in a ring.

For the ring $$A=\mathbb{Z}[i\sqrt{3}]=\{a+i\sqrt{3}b:a,b\in \mathbb{Z}\}$$ I had to show that the only invertible elements are $1$ and $-1$, using the norm $$N:\mathbb{Z}[i\sqrt{3}]\quad ...
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57 views

Check my answer - Finding the jacobi matrix of a function

We are given $f: \mathbb R^n \to \mathbb R^n$ such that: $0 \neq x \in \mathbb R^n$, $f(x)=\frac{x}{|x|}$, where $|x| = \sqrt {x_1^2 +x_2^2+...+x_n^2}$ Find the jacobi matrix (the differential ...
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130 views

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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109 views

Computing H1 norm numerically

I'm solving a PDE numerically using FDM and Spectral Methods. I understand how to compute the $L_{2}$, but I dont understand how to compute the $H_{1}$ norm. What does the $u'$ mean in the below ...
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71 views

question about norms and convex set

Suppose $\overline{B}(0;1) = \{ x \in X : ||x|| \leq 1 \}$ is the closed unit ball on a vector space $X$. MY question is: is the following true? If $\overline{B}(0,1) $ is not convex, then $|| \cdot ...
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23 views

Verifying whether a given function can be a norm.

I was asked to prove that given the vector space $\Bbb{R}\times\Bbb{R}$, the function $f(p)=(\sqrt{a}+\sqrt{b})^2$, where $p=(a,b)$, does not define a norm (on $\Bbb{R}\times\Bbb{R}$). Is the ...
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119 views

About $f(s)=\sum_{a^2+b^2>0} \frac{(-1)^{a^2+b^2}}{(a^2+b^2)^s}=0$ and the Extended Riemann Hypothesis.

Let $s$ be a complex number with a strictly positive real part ($Re(s)>0$). Let $f(s)=\sum_{a^2+b^2>0} \dfrac{(-1)^{a^2+b^2}}{(a^2+b^2)^s}$ where the sum runs over all positive integers $a,b$ ...
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45 views

Solving a linear matrix equation with respect to the maximum of the euclidian distances between rows.

With $n>m$, real number matrices $A$, $B$, $C$ are shaped like: $$A=\left( \begin{array}{ccc} a_{1,1} & \cdots & a_{1,m} \\ \vdots & \ddots & \vdots \\ a_{n,m} & \cdots ...
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76 views

For general non-symmetric square matrices is there a matrix norm that is invariant under similarity transformations?

I think that there is no similarity-invariant matrix norm for general matrices. But are there similarity invariant norms for special types of matrices (e.g. for matrices whose eigevalues are different ...
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62 views

What is the correct notation for defining norms in measure spaces?

For a function $f \in L_2(R)$, we can define its norm as $$ \|f\|_2^2 = \int f^2(x) dx $$ If I use a different measure $\mu$, I can in turn define the norm as $$ \|f\|_2^2 = \int f^2(x) \mu (x) dx ...
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94 views

addition and multiplication of functions in function space, continuous?

I have a norm that works in function space of C[0,1]. How do I show that addition and multiplication of functions (C[0,1]xC[0,1]->C[0,1]) are continuous functions?
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50 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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270 views

Minimum L1 norm may not obtain the sparsest solution?

Here is a paper called For Most Large Underdetermined Systems of Equations, the Minimal L1-norm Near-Solution Approximates the Sparsest Near-Solution. However, I did not quite get its definition of ...
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95 views

How to solve an optimization problem with non-convex Frobenius norm constraint?

The form of my problem is: $$ \min_W \|Y-WX\|_F^2-\|V-WU\|_F^2 $$ $$ s.t. \|W\|_F=1 $$ All five variables are matrices. Since the norm constraint is a non-convex one, I have no idea how to solve this ...
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65 views

Equivalence of given norms in $\mathbb{R}^k$

Find as big as possible $c>0$ and as small as possible $C>0$ such that we have: $$\forall_{\vec{x}\in\mathbb{R}^k} \ \ c\cdot \sqrt{\sum_{j=1}^k|x_j|^2}\le \sqrt{\sum_{j=1}^k ...
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68 views

inequality about Inner product and norm

If $m\times n~(m<n)$ matrix $A$ satisfy the following condition $(1-\delta)||s||_2^2\leqslant \|As\|^2_2\leqslant (1+\delta)\|s\|_2^2$ for all the $n \times 1$ vector with no more than $k$ nonzero ...
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30 views

$|Av||A^{-1}v|$, $A$ non-singular, $|v|=1$.

Let $A$ a non-singular $n \times n$ matrix, $v \in \mathbb{R}^{n}$ a variable vector. The operator norm of $A$ is defined to be $|| A ||=\max_{|v|=1} |Av|$ where $|Av|$ is the standard Euclidean norm ...
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78 views

Hölder Space Definition

At the beginiing of the defintion of Hölder spaces and the Hölder space norm. They start defining the first term of the Hölder norm as follows: If $u:U \rightarrow \mathbb{R}$ is bounded and ...
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121 views

Triangle inequality for $\sqrt{x^T\cdot A\cdot x}$

I want to prove that $f(x):= \sqrt{x^TAx}$ is a norm, $A \in \mathbb{R}^{n \times n}$ positive definite, $x \in \mathbb{R}^n$. I already proved its positivity and absolute homogeneity, but I don't ...
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54 views

Hints on calculating an integral

Suppose $x(t)$ and $y(t')$ are curves traversing the boundary of $[0,1]^2$ in $R^2$ counterclockwise. What is the integral of the following: $$\int\limits_{t,t'}{\frac{dt\,dt'}{\|x(t)-y(t')\|}}$$ I ...
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Given $\mathbf{A}$ stable (all negative eigenvalues), produce a bound on $\|\mathbf{B}\|$ such that…

Given a system: $\dot{\mathbf{x}}=\left(\mathbf{A}+\mathbf{B}\right)\mathbf{x}$ Can you bound $\|\mathbf{B}\|$ s.t. the origin of $\mathbf{x}$ is exponentially stable using a Lyapunov function? ...
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51 views

How to solve linear system of equations in 1 and inf-norm?

I have the problem to find a linear program that is equivalent to solving the problem that finds a minimum for $||Ax-b||_1$ and $||Ax-b||_{\infty}$. We defined a linear program as follows: $min_{x} ...
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24 views

Compute $\|A(t)\|$

When I do my homework (stability theory), I must use the knowledge to the norm of matrix. But I don't remember it (I mean that I'm not sure). My problem: For matrix $A(t)$ is continuous. Compute ...
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Matrix Norms Inequality Proof

How do I prove this inequality / equivalence about matrix p-norms? It appears on the wikipedia and mathworld.wolfram pages on matrix norms without proof. $\|A\|^2_2 \leq \|A\|_1 \|A|\|_\infty$ Maybe ...
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136 views

Grid Function Norm

I am studying about the global truncation error in finite difference methods and I have a question about calculating the error in a Boundary Value Problem (BVP). If we take a simple 1-D problem, the ...
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92 views

What is an operator norm?

I came across this symbol where they are mentioning operator norms. I am not sure how it is different from the Frobenius norm. It was something like this: $|||\Omega-\hat{\Omega} |||_2$ where ...
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141 views

Convergence of the Jacobi iteration method

I think I am not quite understanding the Jacobi Method or some related concept for indirectly solving linear systems of equations of the form $Ax=b$. We need the norm $||I-Q^{-1}A||_\infty < 1$ and ...
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77 views

Distance of a point from a subspace vs. diameter

Let X = $(\Bbb R^N, \|\cdot\|)$ be a Banach space. Let $x_0 \in S^{N-1} = \{x \in \Bbb R^N : \sqrt{x_1^2+...+x_n^2}=1\}$. Denote $B^N_2 = \{x \in \Bbb R^N : \sqrt{x_1^2+...+x_N^2} \le 1 \}$. Define ...
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27 views

discrete version of $L_p$

Why is $(\sum_{i=1}^{N-1}e^{-px_i/\epsilon}\bar{h}_i)^{1/p}=O(N^{-1/p})$,as $N$ approaches 0, where $\bar{h}_i=(h_{i+1}+h_i)/2$, $h_i=x_i-x_{i-1}$? The integral version is much easier to calculate.
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101 views

Prove that: $\|f\|$ is constant in $G$.

Suppose $G$ is a connected open set of $E$ and $f \in \mathcal{H}(G,F)= \{f: G \to F$, $f$ $\text{is holomorphic mapping} \}$. Suppose there is a points $a \in G$, such that $\|f(x)\| \le \|f(a)\|$, ...
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69 views

Equality of expressions containing supremum of some double sums

I am doing a project related to operator norm and some double sequences. In the course of proving some results, I encounter the following expressions: $\displaystyle \|\alpha\| := \sup_{\|x\|_p=1} ...
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31 views

Absolute summable sequence in normed space [duplicate]

I have studied that for a sequence of real numbers absolute summability implies summability. What can we say about the sequences $\{x_k\}$ in a normed space . If it is not true in general could ...
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207 views

Proof of non-strictly convexity of $l_1$ and $l_{\infty}$

Given the Banach space $l_p = \left\{f\in R^n : \|f\| \leq \infty\right\}$ for $1\leq p \leq \infty$, we define the following norms $\|f\|_p = \left(\displaystyle\sum_{j=0}^{\infty} ...
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464 views

How to expand equation inside the L2-norm?

I want expand an L2-norm with some matrix operation inside. Assume I have a regression $Y=X\beta+\epsilon$. I want to solve (meaning expand), $$\displaystyle\|Y-X\beta \|_{2}^2$$ Should I do: 1) ...
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79 views

Differences between a norm in $\mathbb{C}^n$ and a norm in $\mathbb{R}^n$

Consider the definitions of matrix norm and subordinate matrix norm from Matrix Norm set #2 and let $A$ a real matrix and $\|\cdot\|$ a vector norm over $\mathbb{C}^n$. Define \begin{eqnarray*} ...
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69 views

For any linearly independent vectors $x, y$, there is such a norm that $||x||_*>||y||_*$

Today I've seen in my class that: For any linearly independent vectors $x, y$, there is such a norm that $||x||_*>||y||_*$ Our lecturer called it Benchmark theorem. I wanted to learn more ...
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263 views

Gradient with respect to a matrix variable

I want to find the gradient of the function $\mathcal{F}_1$ with respect to the matrix $\mathbf{X}$ (differentiate with respect to $\mathbf{X}$): $$ \mathcal{F}_1 (\mathbf{X}; \mathbf{\lambda})= ...
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46 views

A modular which is not a metrizing modular (hence not an F-norm)?

I'm taking the terminology from Rolewicz's 1985 Metric Linear Spaces. Given a complex vector space $X$, a modular $m$ is any function $m:X\to[0,+\infty]$ satisfying the following for all $x,y\in X$ ...
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102 views

Why is the negative entropy Lipschitz with respect to the $1$-norm (Over)?

Let $\left\|x \right\| = \sum_{i=1}^{i=n}\left|x^i\right|$ and $d\left(x\right)=\sum_{i=1}^{i=n}x^i\ln x^i$ where $x\in R^n $ and $ \sum_{i=1}^{i=n}x^i=1$ How to prove: For all $x, x'$, $$\left| ...
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113 views

Hölder norm estimates

How do you prove the following estimate for composition of functions: If $k\geq 1$, then there exists a constant $c=c(k,\alpha)$ such that $$ \|f_1\circ g_1-f_2\circ g_2\|_{k,\alpha}\leq ...
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60 views

Optimizing over norms of set of equations.

I have a set of N real-valued finite-dimensional vectors $\mathbf{v}_i$ and target norms $y_i$ and I am trying to find a linear transformation matrix $L$ such that the norm of the transformed vectors ...
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247 views

Dual norm and distance

Let $Z$ be a subspace of a normed linear space $X$ and $x\in X$ has distance $d=\inf\{||z-y||:z\in Z\}$ to $Z$. I would like to find a function $f\in X^*$ that satifies $||f||\le1$, $f(x)=d$ and ...
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275 views

Why is the subdifferential of norm of a matrix ||A|| defined like this?

I read in a paper called "Characterization of the subdifferential of some matrix norms" that it defines the subdifferential of the matrix norm like this: $$\partial ||A||=\{G \in R^{m\times n} : ...