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

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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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142 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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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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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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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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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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70 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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32 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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2answers
225 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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514 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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73 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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1answer
271 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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104 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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114 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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61 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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1answer
248 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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286 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} : ...
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1answer
110 views

Matrix norm relationship

Suppose we have two matrices, A and B, and $\left\Vert A\right\Vert _{F}\geq\left\Vert B\right\Vert _{F}$ where $\left\Vert .\right\Vert _{F}$ denotes Frobenius norm. Does it imply $\left\Vert ...
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39 views

Prove that $\|{e^{At}x_o}\| \geq e^{-\lambda t}\|{x_o}\|$

Prove that $\|{e^{At}x_o}\| \geq e^{-\lambda t}\|{x_o}\|$, for some $\lambda \gt 0$, A is $n\times n$ matrix and $x_o$ is a $n \times 1$ vector.
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1answer
38 views

How to implement fuzzy minimum function via fuzzy maximum

I know that I can represent fuzzy max via power function(i need it in neural network) i.e. def max(p:Double)(a:Double,b:Double) = pow(pow(a,p) + pow(b,p) , 1/p) // assumption a >=0 and b >=0 ...
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246 views

Bound on euclidean norm

Is it possible to find a suitable lower bound on $$\left|\left(tM+\sum_{k=2}^\infty\frac{(tM)^k}{k!}\right)\cdot b\right|$$ for $M$ as $n \times n$ matrix, $b$ as $1 \times n$ vector and for all $t$ ...
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104 views

How to prove an inequality for a special structure of strictly triangular matrix

The problem I cause is attached below. I am trying to prove the inequality. By using small $M$, I found that the terms on the left side of the inequality are part of the terms of the expansion on ...
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329 views

Dual Norm Of Sum Of Norms

What is the dual of a norm that is the sum of two-norms? Specifically, say we have the following norm for a $\mathbf{x}\in \mathbb{R}^n$ and $\mathbf{A}_i \in \mathbb{R}^{m \times n}$ $\|\mathbf{x}\| ...
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103 views

Inequality estimation

Let $B$ be an open unit ball in $\mathbb{R}^d$ centered at the origin and $u$ be a twice continuously differentiable function on $\bar{B}$ with $u|_{\partial B} = 0$. Know $$\Delta u = f.$$ How can I ...
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325 views

Prove inequality with norms and matrices

Prove that if $A$ is invertible and $||A-B||<||A^{-1}||^{-1}$ then $$\lVert A^{-1} - B^{-1}\rVert \leq \lVert A^{-1}\rVert \frac{\lVert I-A^{-1}B\rVert}{1-\lVert I-A^{-1}B\rVert}.$$ I also need ...
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constrained optimization of dot product

Given a real matrix $A$ find a positive vector $x$ of unit length ($x^T x = 1$) for which $x^T A^T A x$ is minimal (closest to $0$). A has size about $1000 \times 20$ and can be written as $[ A_P | ...
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113 views

How to define an inner product for normed space? [duplicate]

Possible Duplicates: Norms Induced by Inner Products From norm to scalar product Given a normed space $(X, \|\cdot\|)$. Is it possible to define an inner product $\langle \cdot, \cdot ...
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130 views

Show that $A^{n}\to0$ if and only if $\|A\|^{n}\to0$

Let $\boldsymbol{A}$ be a square matrix. Show that $\lim_{n\to\infty}\boldsymbol{A}^{n}=0$ if and only if $\lim_{n\to\infty}\|\boldsymbol{A}\|^{n}=0$ for the spectral radius or for some operator ...
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1answer
48 views

Is it possible that $\Vert I \Vert > 1$ !?

For any matrix norm, is it possible $\Vert I \Vert > 1$ ?, where $I_{n\times n}$ is identity matrix. If not, why in some books they write $\Vert I \Vert \geqslant 1 $ ?
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3answers
70 views

About matrix products $A^{T}A$ and $ AA^{T} $

I'm investigating the relationship between 2-norms and eigenvalues of $A^{T}A$ and $ AA^{T} $, in order to better understand the SVD decomposition. How can I prove that $A^{T}A$ and $ AA^{T} $ are ...
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6answers
77 views

Why does this set a compact?

Consider the following set in $\mathbb{R}^n$: $$S = \{ x \ :\ \|x\| = 1 \}$$ Why is this set a compact? I don't know about closed, but it doesn't look bounded to me.
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Topology induced by norm

What is the meaning of topology induced by norm. To me topology is a collection of subsets satisfying certain rules. How can a norm induce a topology...? For example how can $\|\cdot\|_{2}$ induce a ...
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how to interpret this norm

Could any one tell me what "x" is there when he has defined $\|x\|$, just after he says $M_n$ has the operator norm thank you for helping.
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139 views

Is it true that $\biggl\|I-\frac{vv^T}{v^Tv}\biggr\|=1$?

I am following a proof in the text OPTIMIZATION THEORY AND METHODS a springer series by WENYU SUN and YA-XIANG YUAN. I come across what seems obvious that for a column vector $v$, with dimension ...
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Is convergence in the norm equivalent to convergence of norms?

If $\| \cdot \|$ is a norm on some space. Does the equivalence go both ways? $$\| f_n-f \| \to 0 \Longleftrightarrow \| f_n\| \to \| f\| $$ The $\implies$ direction is obvious since $\| f_n-f \| ...
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156 views

Prove $\|f\|_{L^p}$ is not equivalent to $\|f\|_{\infty}$ in $C[a,b]$

Prove that in $C[a,b]$ the uniform norm is not equivalent to the $L^p$ norm for $(1\leq p < \infty)$ I am stuck on showing that the function below satifies the claim. I know that f is continuous ...
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Difficulty in understanding a step in a definition in the book Walter Rudin

In the book Principles of mathematical analysis by Walter Rudin,He writes: "For $ A\in L(\Bbb R^n,\Bbb R^m)$, define the norm $||A||$ of $A$ to be sup of all numbers $|Ax|$, where $x$ ranges over all ...
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Why $||f||=\mathrm{sup}_{||g||=1}|\langle{f,g}\rangle|$

My question is why $||f||=\mathrm{sup}_{||g||=1}|\langle{f,g}\rangle|$? for any two functions $f$ and $g$ with $||g||=1$, and $||\;||$ denotes the 2-norm. I have tried to use the triangle inequality ...
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68 views

Norm of vectors inequality

I tried proving this with triangular inequality but i was not right can any one help me with this
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Mysterious Matrix Norm

Given a matrix $M$, does anyone know the name and the definition of the following norm? $$ \|M\|_* $$ Thanks in advance, Francesco.
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Norm on vector space.

I was given a true or false questionnaire to study for my final and do not know if I am right or wrong about these statements. I marked the following statement as True: If $\|\cdot\|$ is a norm on ...
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1answer
857 views

Prove that $\|T\|=\sup\{\|T(x)\|: x\in X , \|x\|<1 \}$ for $T$ a bounded linear operator

Let $T: X \to Y$ a bounded linear operator. Prove that $$\|T\|=\sup\{\|T(x)\|: x\in X , \|x\|<1 \}.$$ It is $||T||= \sup \{||T(x)||: x\in X ,||x|| \leq1 \}$ so $||T|| \geq \sup\{||T(x)||: x\in X , ...
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Cauchy-Schwarz, how does it work in this example?

Consider $\|\cdot\|_2$ such that $\|x\|_2 = \left(\sum_{i=1}^n |x_i|^2\right)^{1/2}$. Let $A \in \mathbb{R}^{n\times n}, x\in \mathbb{R}^n$, then $$\begin{align} \|Ax\|_2^2 & = \sum_{i=1}^n ...
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37 views

norms of Symmetric Positive Definite Submatrices

Let $\mathbf{A}\in\mathbb{R}^{N \times N}$ be symmetric positive definite. For some $1\leq k<N$, partition $$\mathbf{A}=\begin{pmatrix}\mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{12}^T ...
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1answer
58 views

Cauchy Schwarz inequality and absolute value

Here's the inequality: $$|\langle u,v\rangle|\le\|u\|\cdot\|v\| $$ Why on the LHS there's an absolute value? We know that $\langle u,v\rangle \ge 0$ Isn't it redundant?
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278 views

symbol for maximum number in an array or vector

As we know $\|x\|_{\infty}$ returns the maximum "absolute" value through the vector . I'm looking for a symbol to show the actual value that can be also negative: $ ?x?=\max\{x_1,...x_N\}$
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49 views

Does $d(x,y) = \lvert N(x) - N(y)\rvert$ satisfy the triangular inequality?

Let $N(x)$ be the norm of the vector $X$ and efine $$d(x,y) = |N(x) - N(y)|$$ I want to prove that $d(x,y)$ satisfies the triangular inequality. Here is my attempt: $$|N(x) - N(y)| \leq |N(x)| + ...