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 the Linear (Integral) Operator on $C[0,1]$

There may be some similar questions being asked but I didn't see anything that was quite what I was looking for so I'll ask the question here. Let $X = (C[0,1],||.||_{\max})$ and $T:X \to X = ...
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Help proof regarding sequence in subset of Hilbert space

I'm to prove the following: Let $H$ be a Hilbert space, and let $M$ be a non-empty convex subset of $H$. Suppose that $(x_n)$ is a sequence in $M$ such that $ ||x_n|| \to d$, where $d= \inf ...
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Is a minimum a minimum in every norm? (for finite-dimensional vector spaces)

I have a question regarding the equivalence of the norms in finite-dimensional vector spaces. Basically the question is: if $\hat{x}$ is some minimum-norm solution in a subspace $\mathcal{K}$ under ...
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103 views

Condition number - proof

I have a problem with which I've been struggling for a while. It's probably not that difficult, but I seem to be stuck so... here we are. Let A be an invertible lower- or upper-triangular matrix ...
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Norm on a Geometric Algebra

In the literature, for example "New Foundations for Classical Mechanics" by David Hestenes, the author introduces a function on the Geometric Algebra $$||M||^2=\langle M M^\dagger \rangle_0,$$ where ...
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Norm preserving Matrix properties

Norm-2 preserving can be done using unitary/orthogonal matrix: $A^*A = I => ||Ax|| = ||x||$ What is the matrix other than identity matrix that can preserve other norms ( norm-1, norm-inf) ?
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Norm equivalence (Frobenius and infinity)

I am trying to prove the matrix norm equivalence for norms 1, 2, $\infty$ and Frobenius. I have managed to solve find the constants for $||.||_{1}$ and $||.||_{2}$ but I cannot see how to continue ...
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Norm of the one dimension real space

I read that apart from a constant factor the absolute value is the only norm on the vector space $\mathbb{R}^1$. Why is that so?
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Lipschitz continuity and gradient of a real-valued function on a normed space

The gradient of a function $f:E\to\mathbb{R}$ is Lipschitz continuous with parameter $L > 0$ iff $$\|\nabla f(x) - \nabla f(y)\|^* \le L\|x-y\| \quad \forall x,y\in E.$$ I have two questions: ...
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435 views

Frobenius norm of a matrix [closed]

I know that Frobenius norm of a matrix A is equal to the square root of the trace of (A*conjugate transpose(A)). But how do I prove it mathematically?
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Dual Optimization Problem

I have the following optimization problem, $$ \text{minimize}_{X,Y} \ \lVert X\rVert_* + \lambda \lVert Y\rVert_1 \\ \text{subject to }X + Y = C$$ $C \in \mathcal{R}^{m \times n}$, $\lVert ...
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Are there norms on $\Bbb{C}^m$ and $\Bbb{C}^n$ so that the norm $\Vert\cdot\Vert$ is a subordinate norm?

Denote $$\Vert A\Vert=\sum_{1\le j,k\le m}\vert A_{j,k}\vert$$ is cleary a norm over $M_{m,n}(\Bbb{C})$ but not a subordinate norm by taking the identity matrix $I$. So my question is: Can we make ...
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52 views

Is this a matrix operator norm?

Is the max element-wise norm a matrix operator norm? I know a matrix operator norm is defined by $$ |A|_p=\sup_{v≠0} \frac{|Av|_p}{|v|_p} $$ But how can I tell if the max norm is an operator norm?
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Finding the norm of a linear functional

This is a basic question of functional analysis, but I want to know how to... Find the norm of the linear functional $f$ defined on $C[-1,1]$ by $$f(x)=\int_{-1}^0 x(t) \, dt - \int_0^1 x(t) \, ...
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Inequality for subordinate norm of a submatrix : $\Vert B\Vert\le\Vert A\Vert$.

We 'did' this exercise in class Show that the subordinate norm of a submatrix is smaller than the the subordinate norm of the matrix i.e. $$\Vert B\Vert\le\Vert A\Vert$$ where $B$ is the sub ...
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Inequality $\vert x^{*}y\vert\le \Vert x\Vert_1\Vert y\Vert_{\infty}.$

I would like to prove that the following exercise : For all $x,y\in \Bbb{K}^n$ we have $$\vert x^{*}y\vert\le \Vert x\Vert_1\Vert y\Vert_{\infty}.$$ Where $\Bbb{K}=\Bbb{R}$ or $\Bbb{C}$ and ...
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Subordinate norm is equal to $\Vert A\Vert_M=\Vert MAM^{-1}\Vert$ for the norm $\Vert x\Vert_M=\Vert Mx\Vert$

Let $M\in \mathcal{M}_n(\Bbb{K})$ a nonsingular matrix, we have $\Vert x\Vert_M=\Vert Mx\Vert$ is a norm over $\Bbb{K}^n$. I have to prove that the subordinate norm is equal to $\Vert A\Vert_M=\Vert ...
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Proof of Matrix Norm (Inverse Matrix)

Show for any induced matrix norm and nonsingular matrix A that $$ \left\|A^{-1}\right\| ≥ (\left\|A\right\|)^{-1} $$ where $$ \left\|A^{-1}\right\| = ...
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73 views

Show that $\infty$-norm and $C^1$-norm are not equivalent.

Show that $\infty$-norm and $C^1$-norm are not equivalent. For the $C^1([a,b],\mathbb{R})$ space, show that $\displaystyle ||g||_\infty=sup_{a\leq t\leq b}|g(t)|$ and $\displaystyle ...
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Matrix norm square properties.

I'm trying to prove one of these inequalities. This isn't a homework problem but trying to solve out of curiosity as it didn't have any relationship between $x$ and $\alpha$. How do you prove: ...
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31 views

Norm2 of a vector of complex numbers

I am migrating a matlab code into C++ and I need to know how does matlab calculate the norm of below matrix. For two numbers, A=a+ib , B=c+id, I know I should do [(a-c)^2+(b-d)^2]^1/2. But how is it ...
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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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How to prove that the L2 norm is a non-increasing function of time for a 2nd-order PDE?

I am having a test in few days and I saw an interesting question while I was skimming through the book problems. The problem is concerned about initial-boundary value problem of 2nd order PDEs. To ...
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Operator norm and Hilbert Schmidt norm

I'm looking for a proof of \begin{equation} ||T||\leq ||T||_{HS}, \end{equation} for which it is sufficient to show \begin{equation} ||Tx|| \leq ||x|| \cdot ||T||_{HS} \forall x\in H, x\not=0 ...
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50 views

An inequality using Sobolev norms

Let $\| \cdot \|_{H^s(\mathbb R)}$ be the usual Sobolev norm in $\mathbb R$ and $r>0$. If we have $$ \|f\|_{L^\infty(\mathbb R)} \le \| f\|_{H^k(\mathbb R)} $$ for all $k>r$, then the inequality ...
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29 views

Distance of two vectors under L_inf norm

Very simple question: suppose I have two vectors $a = (1,-2)$ and $b = (4,2)$. Under the L_inf norm, would the distance between them be $abs( ||a||_{inf} - ||b||_{inf}) = abs(2 - 4) = 2$? Is this the ...
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Norm of an integral operator

I have an exercise that I need to solve and I can't finish it. Let $k \in \mathcal{C}([0,1] \to \mathbb{R})$. Proove that this operator : $$ \begin{array}{ccccc} T & : & ...
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Can we equip smooth functions ($C^{\infty}[0,1]$) to some complete norm? [duplicate]

Let $C^{\infty}[0,1]$ be the vector space of all real function $f:[0,1]\rightarrow \Bbb R$ s.t for each $n \in \Bbb N$, $f^{(n)}$ exist. Is there a norm $\|.\|$ on $C^{\infty}[0,1]$ such that ...
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The supremum norm is submultiplicative

Is the following proof correct: Let $X$ be compact a compact Hausdorff space and $C(X)$ the continuous functions $f: X \to \mathbb{C}$ on X. We can equip $C(X)$ with the (edit: sorry, semi-)norm ...
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Concepts of isomorphisms of linear spaces with a norm and inner product

If I have a topological space, I say that a homeomorphic map preserves the structure of this space. Thus, in order to preserve topological properties we want to have a continuous bijection with a ...
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Showing that a map $x \to \|x\|$ is continuous?

I am given this: Consider a real Banach space $X$ with norm $\|*\|$. 1) Show that the map $x\to \|x\|$ from $X$ to $\mathbb{R}$ is continuous. Is it uniformly continuous? 2) Show that the maps ...
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What is a good reference for learning about induced norms?

Wikipedia tells me a little about it. Following the wiki-link treasure hunt leads me to topics such as "p-norms on finite dimensional vector spaces". Which makes me want to ask: what's a good ...
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question about “Norm”

i have this space $H=\lbrace u\in AC([0,+\infty), u(0)=u(+\infty)=0, \sqrt{p}u'\in L^2((0,+\infty))\rbrace$ where $p>0$ and $\displaystyle\frac1p\in L^1$ how to see that the quantity: ...
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43 views

Find the coefficients $a,b$ so that $\|\vec x\|=a(|x_1|+|x_2|)+b\max(|x_1|,|x_2|)$ is a norm

For which coefficients $a,b$, the expression: $$\|\vec x\|=a(|x_1|+|x_2|)+b\max(|x_1|,|x_2|)$$ is a norm in $\mathbb R^2$? My attempt: I need to verify the properties of the norm: Triangle ...
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Shrinkage operator for matrices

Here http://web.stanford.edu/~boyd/papers/pdf/prox_algs.pdf, on page 188, you can see the derivation of the soft thresholding operator or shrinkage operator for the case of vectors using Moreau ...
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Frobenius norm and Gaussian noise

Why Frobenius norm is considered to a good tool for dealing with Gaussian noise?
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Any two points inside a circle are within a diameter of each other.

In many problems involving the Pigeonhole Principle, we often assume the following lemma: Lemma: The distance between any two points in a circle of radius $r$ is at most $2r$. Intuitively, this ...
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proof about real sequences

Let $V$ be the space of real sequences {${x_{1},x_{2},...}$} so that $\sum_{k=1}^{\infty}x_{k}^{2}$ converges. Let $W\subset V$ be the set of rational sequences with a finite number of terms. ...
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Prove that Frobenius matrix norm is compatible with the vector norm

Show that, the Frobenius matrix norm $||.||_F$ is compatible or consistent with a vector norm $||.||_2$ , that is, $||Ax||_2 \leq ||A||_F ||x||_2, \forall x \in \mathbb{R}^n$. Where $||A||_F = \sqrt{ ...
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geometric interpretation of the norm: $\|\vec x\|={(|x_1|+|x_2|)\over 3}+{2\max(|x_1|,|x_2|)\over 3}$

Let $p:\mathbb R^2 \to \mathbb R$ be a norm so that $$\|\vec x\|={(|x_1|+|x_2|)\over 3}+{2\max(|x_1|,|x_2|)\over 3}$$= $${{\|\vec x\|_1\over 3}}+{2\|\vec x\|_\infty\over 3}$$ The thing is that I need ...
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The meaning of notation $\|x - x^*\|$

I was just wondering what $\|x - x^*\|$ in the following equation means: $$B(\epsilon) = \{x : \|x - x^*\|<\epsilon\} $$ Thanks.
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Dual norm of the matrix $L^1$ norm is infinity norm (and vice versa)

Recall that for a given norm $\|\cdot\|$ on $\mathbb{R}^n$, the dual norm is defined as a function $\|\cdot\|_*: \mathbb{R}^n \rightarrow \mathbb{R}$ with: $\|y\|_* = \max \limits_x \{x^Ty: \|x\|\le1 ...
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Proving result on spectral radius

How do I prove that $$\rho(A)=\inf\limits_{\text{operator norms}}\|A\|,$$ $\rho$ being the spectral radius, $A$ being a complex $n\times n$ matrix and operator norms being induced from vector norms by ...
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Is Frobenius norm induced up to a scalar factor?

I know that the Frobenius norm is not induced since $||I||_F=\sqrt n\neq 1$. But what if we consider the norm $\frac 1 {\sqrt n} ||\cdot ||_F$? Thank you!
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Product of Norm of Vectors

I have a simple question ... Consider two vectors of same length $\mathbf{x}$ and $\mathbf{y}$. This identity holds: $|\mathbf{x}'\mathbf{y}|$ $\leq$ $\|\mathbf{x}\|$ $\|\mathbf{y}\|$ Then is ...
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Why is semi-norm special?

One difference between semi-norm and norm is: "It is possible for $\|v\| = 0$ for nonzero v, $\|\cdot\|$ being semi-norm" I see some papers, and they use semi-norm directly. Why is semi-norm ...
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Dual norm intuition

The dual of a norm $\|\cdot \|$ is defined as: $$\|z\|_* = \sup \{ z^Tx \text{ } | \text{ } \|x\| \le 1\}$$ Could anybody give me an intuition of this concept? I know the definition, I am using it ...
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115 views

When does the equality hold for norm equivalence

We know that for a vector $x\in \mathbb{R}^n$, its 1-norm and 2-norm satisfy that $$\frac{1}{n}\|x\|_1\le\|x\|_2\le \|x\|_1,$$ could anyone please give me some hints that on what condition these ...
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Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} ...
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90 views

Proving result on matrix norms

How do I prove that, letting, for $A\in\mathbb{C}^{n\times n}$: $$(a)\quad\|A\|_1=\max\limits_{i=1,\cdots,n}\left(\sum\limits_{j=1}^n|a_{ij}|\right),$$ $$(b)\quad\|A\|_2=\rho^{\frac12}(A^HA),$$ with ...