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

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whats the difference between $|v|$ and $||v||$?

$v$ being a vector. I never understood what they mean and haven't found online resources. Just a quick question. Thought it was absolute and magnitude respectively when regarding vectors. need ...
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Example of two norms on same space, non-equivalent, with one dominating the other

I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other...
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88 views

Does Permuting the Rows of a Matrix $A$ Change the Absolute Row Sum of $A^{-1}$?

For $A = (a_{ij})$ an $n \times n$ matrix, the absolute row sum of $A$ is $$ \|A\|_{\infty} = \max_{1 \leq i \leq n} \sum_{j=1}^{n} |a_{ij}|. $$ Let $A$ be a given $n \times n$ matrix and let $A_0$ ...
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454 views

Frobenius norm is not induced

My textbook says: Frobenius norm defined on $\mathbb{R}^{m,n}$ by a formula: $$\|A\|_F=\sqrt{\sum_{i=1}^n\sum_{j=1}^m|a_{i,j}|^2}$$ when $n>1, \ m>1$ is not induced by any vector's ...
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Norm of Hilbert matrix is it equal to $\pi$?

Let $A$ be a Hilbert matrix, $$a_{ij}=\frac{1}{1+i+j}$$ We have the following result : $\Vert A\Vert\leq \pi$. I am using the subordinate norm of the euclidean norm i.e. $$ \Vert A\Vert=\sup\{\langle ...
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175 views

Is $l^2$ norm differentiable at $x=0$?

For $l^2$ norm on $\mathbb{R}^d$, $\frac{d\|x\|_2}{dx} = \frac{x}{\|x\|_2} $, so $d\|x\|_2$ is differentiable wherever $x \neq 0$. is the norm differentiable at $x=0 \in \mathbb R^d$? Thanks!
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Norm of differentiation operator $Tf(t)=f^{'}$..

Consider $T:C^1[0, 1]\rightarrow C[0, 1]$ given by $Tf=f'$ where $$\|f\|_{C^1}=\|f\|_\infty+\|f'\|_\infty$$ and $\|f\|_\infty=\sup_{x\in [0, 1]}|f(x)|$. How to prove $\|T\|=1$? The inequality ...
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137 views

Linear Algebra munkres analysis on manifolds question.

If $A$ is an $n$ by $m$ matrix and $B$ is an $m$ by $p$ matrix, then $$ |AB| \leq m|A||B|$$ where $|A| = \max\{|a_{ij}| : i = 1,\ldots,n \text{ and} j = 1,\ldots,m\}$ Attempt: $ |AB| = \max\{| ...
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188 views

Inequality involving norm and inner product

I am stuck proving this trivial inequality: on a real inner product space, $(||x||+||y||)\frac{\langle x,y\rangle}{||x|| \cdot ||y||}\leq||x+y||$ I have tried to square both sides and use the Cauchy ...
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206 views

Poincare Inequality implies Equivalent Norms

I am currently working through the subject of Sobolev Spaces using the book 'Partial Differential Equations' by Lawrence Evans. After the result proving the Poincare Inequality it says the following ...
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99 views

When does $\|x+y\|=\|x\|+\|y\|?$

Let $(V_\mathbb R,\langle,\rangle)$ be an inner product space. I'm trying to see for $x,y\in V$ when does $\|x+y\|=\|x\|+\|y\|?$ Let $\|x+y\|=\|x\|+\|y\|$ Squaring both sides, $\langle ...
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158 views

The trace norm cannot be increased by composing with a unitary operator

$\newcommand{\tr}{\operatorname{tr}}$ I was reading a proof for the statement $|\tr(US)|\leq |\tr(S)|$, for every endomorphism $S$ on a complex vector space $H$ and every unitary operator $U$ on the ...
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125 views

Geometric Sums in Banach Algebra

Let $E$ be a Banach Algebra with identity, and $v\in E$, so that $||v|| < 1$. The geometric series $w = \sum_{k=0}^\infty v^k$ converges in the norm. I can show that $||w|| \le ...
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35 views

Is the trace of a matrix a norm?

If the matrix norm of A is defined as $\|A\|=\sum_{i,j}|Aij|$ then how do I determine if the sum of the diagonal elements, i.e., the trace is a valid norm? I am not really sure how to approach this ...
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74 views

Is $\|\nabla u\|_2$ a norm on $W^{1,2}(\mathbb{R}^N)$

Is $\|\nabla u\|_2$ a norm on $W^{1,2}(\mathbb{R}^N)$? I know when $\Omega$ is bounded, $\|\nabla u \|_2$ defines a equivalent norm on $W_0^{1,2}(\Omega)$ with $\|u\|_{1,2}$. And to me it seems ...
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146 views

Norm of Prime Ideal

Show that the norm of a prime ideal in a number field $K$ is a power of some prime number, i.e., if $P$ is a prime ideal in $O_K$ for some number field $K$, then $N_\mathbb{Q}^K(P)=p^n$ for some ...
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120 views

How can we replace $\sup$ with $\max$ in definition of subordinate norm for finite-dimensional vector space?

I have been given the definition of a subordinate (operator or matrix) norm: $$\lvert\lvert A \rvert\rvert=\sup_{x\in V}\frac{\lvert\lvert Ax \rvert\rvert}{\lvert\lvert x \rvert\rvert},$$ where $V$ is ...
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210 views

How is $L_{2}$ Minkowski norm different from $L^{2}$ norm?

I am reading the book Multidimensional Particle Swarm Optimization for Machine Learning and Pattern Recognition. They use $L_{2}$ Minkowski norm (Euclidean) as the distance metric in the feature space ...
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Upper bound for norm of Hilbert space operator

It is a standard result that for a bounded self-adjoint operator $T$ on a complex Hilbert space $H$, we have $||T||=\sup_{||x||=1}|\langle Tx,x\rangle|:=M$. It seems that for any bounded operator on ...
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62 views

Norms of eigenvalues bigger than 1 implies $|Ax|>x$ for all nonzero $x$?

If all the eigenvalues of $A$ (an n by n real matrix) have norms bigger than 1, is it true that $|Ax|>|x|$ for all nonzero $x\in\mathbb{R}^n$? (This is clearly true if $x$ is an eigenvector ...
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72 views

Showing that the zero vector has norm zero

I need to show that this is a property of a norm. I know this is supposed to be straightforward but I am somehow not seeing it. The property is $$\lVert 0\rVert = 0$$ I was trying to use the fact ...
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234 views

Two proofs concerning Hölder's inequality

I am studying functional analysis and I have come across two statements which can be proven by using Hölder's inequality, but I don't know how/why. Reminder: let $a\in l^p$ and $b\in l^q$, then: $$ ...
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318 views

An inequality involving the norms of symmetric positive definite matrices

Given A and B two real symmetric positive definite matrices is it true that, for some norm $\|.\|$, this inequality holds $$ \|AB-I\| \leq \|A^2B^2-I\| \qquad ? $$
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94 views

Bounding the $l_1$ norm of a vector

Let $x$ be real vector with $\|x\|_1=x_1+\ldots +x_{2n}$. How to bound from above $(x_1+\ldots+x_n)(x_{n+1}+\ldots+x_{2n})$ by $l_2$ norm of the vector $x$. Of course, using $\|x\|\leq\sqrt ...
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169 views

Proof that $\|\cdot\|_B$ defines a norm

Can you tell me whether my (partial) proof that $\|v\|_B := \inf \{\lambda > 0 \mid \frac{1}{\lambda} v \in B \}$ where $\varnothing \neq B \subset \mathbb R^d$ is open, bounded, $B = -B$ and ...
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27 views

Definition of the norm of a bounded linear operator.

I have a somewhat basic but confusing question regarding the definition of the norm for bounded linear operator. Suppose $f$ is a bounded linear operator, that is, there exists $M>0$ such that ...
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123 views

Proof that the Euclidean norm is indeed a norm

I apologize beforehand for this question. Its embarrassing I know. Anyway, here we go. Recall: $$ \| x \|^2_{\mathbb{R}^2} = \sum^{n}_{i = 1} x^2_{i}$$ How do we prove its a norm? Well if its a ...
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59 views

Inequality between operator norm and Hilbert-Schmidt norm

I have seen the following inequality here but I don't know where I can find a proof for it. Could somebody give me a hint to understand it or guide me to a reference please? $\|AB\|_{HS} \leq ...
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208 views

Why is semi-norm special?

One difference between semi-norm and norm is: "It is possible for ||v|| = 0 for nonzero v, ||.|| being semi-norm" I see some papers, and they use semi-norm directly. Why is semi-norm better than ...
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172 views

Prove $|\det A| \leq \prod_{j=1}^n ||a_j||$

Let's say A is a square n by n matrix. ||$x$||=$x^T x$ and x is a real n-column norm. How would you show this? I tried to use the QR factorization here in showing that ||$a_j$||=||$r_j$||, but ...
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36 views

Spectral Norm of $2\times 2$ symmetric matrix

Consider a $2\times 2$ symmetric matrix, in this case, is there some closed formula for its spectral norm ? By spectral norm I mean the induced 2-norm, there is a definition here. Thanks.
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Question about sup norm

Let $x \in \mathbb{R}^n$. Define $|x| = \max\{ |x_1|,...,|x_n|\} $. I want to show that this is a norm on $R^n$. This is my reasoning. First, notice $$ |x| = \max\{ |x_i| \} \geq |x_i| \; \forall i ...
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263 views

Prove that the given distance function is a norm

Let the vector space $X=K^3$. For $x=(\alpha_{1}, \alpha_{2}, \alpha_{3}) \in X$, we define $||x||= [(|\alpha_{1}|^2+|\alpha_{2}|^3)^\frac{3}{2} + |\alpha_{3}|^3]^\frac{1}{3}$ Proof that $||·||$ is ...
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144 views

Strange mistake in this proof about normed separable vector spaces.

I am supposed to prove that an infinite-dimensionale separable normed space $X$ constains a countable subset $Y$ that is linearly independent and dense. There are three hints given: First, prove ...
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226 views

Norm of random vector plus constant

Suppose that $w$ is a multivariate standard normal vector and $c$ a real vector of the same size. I know that for positive x $$P(||w+c||^2\geq x)\ \geq \ P(||w||^2\geq x)$$ but I cannot prove it. We ...
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53 views

what is the limit of $l_p$ at p=0?

The p-norm is defined as: $$ \ \|x\|_p=\left(|x_1|^p+|x_2|^p+\dotsb+|x_n|^p\right)^{\frac{1}{p}} $$ When $p<1$, this is no longer a "norm" because it violates the triangle inequality (- it is ...
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229 views

Norm of a linear transformation

Let $T:\mathbb R^2\to \mathbb R^2$ be given by the matrix $\begin{pmatrix}a&b\\ c& d\end{pmatrix}$. Let $u:=a^2+b^2+c^2+d^2+2(ad-bc)$ and $v:=a^2+b^2+c^2+d^2-2(ad-bc)$. I need to show that ...
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189 views

Can the $0$-norm represent determinism?

In Scott Aaronson's Quantum Computing since Democritus, he presents classical probability theory as based on the $1$-norm, and QM as based on the $2$-norm. Call $\{v_1,\ldots,v_N\}$ a unit vector ...
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Do the pth powers of $p$-norms define the same partial ordering on the set of all probability distributions for all $p>1$?

Consider the $p$-th power of the Schatten $p$-norm $||q||_p$ of a probability distribution $q$ , ie, the function $\sum_j q_j^p$, where $\sum_j q_j = 1$ and $q_j \geq 0$. For fixed $q$ and $p>1$ ...
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Is this a matrix norm?

In wikipedia, the operator norm of a matrix is given by (assume: real, $n$-dimensional) $$ ||A||= \max \left\{ \frac{|Ax|}{|x|}:x \in \mathbb{R}^n, x\neq 0 \right\}$$ (I'm not sure why it is not a ...
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148 views

Operator norm and spectrum

Let $L$ be a bounded linear operator on a Hilbert space. Without assuming finite dimensions, can we express the operator norm of $L$ in terms of the spectrum of the positive operator $L^{\dagger}L$? ...
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369 views

Matrix norm inequality implying eigenvector norm inequality

For a matrix $A$ let $\|A\|$ be the norm given by $\|A\|=\sup_{v \neq 0}\frac{\|Av\|}{\|v\|}$ where $\|v\|$ is the Euclidian norm on the vector $v$. Suppose we have matrices $M$ and $S$ with leading ...
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139 views

Bizarre formula for arc length

I'm reading a book on computer science/math and I found this formula for arc lengths that I've not been able to decipher: $$\left|\int_p^q\left\| {df(x)\over dx} \right\| dx\right|$$ where $\lVert ...
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462 views

Is this a norm? (triangle inequality for weighted maximum norm)

I've been trying to prove that the following is a norm, but wasn't successful. I also cannot find a counterexample. So help is greatly appreciated. Let $x \in \mathbb{R}^N, \ w_i \in \mathbb{R}_+,\ ...
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863 views

Proof that every normed vector space is a topological vector space

The topology induced by the norm of a normed vector space is such that the space is a topological vector space. Can you tell me if my proof is correct? Of course we have to show that addition and ...
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127 views

How to prove that a unitary matrix with other conditions is diagonal?

Let $A$ be an $n$ by $n$ complex matrix. The singular value decomposition of $A$ gives: $A=U\Sigma V^*$, where $U$ and $V$ are unitary matrices, and $\Sigma$ is a diagonal matrix that contains the ...
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274 views

What is the norm of the operator $\phi: L^3[-2,2] \to \mathbb{C}$ defined by $ \phi(f)=\int_{0}^{1}e^xf(x-1)dx$?

I'm trying to figure out the norm $\|\phi\|$ of the functional $\phi: L^3[-2,2] \to \mathbb{C}$ defined by $$ \phi(f)=\int_{0}^{1}e^xf(x-1)\mathsf dx$$ but am struggling. I can't figure out how to ...
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25 views

Proving $||x||_{\infty}$ is a norm on $\mathbb{C}^n$

I'm trying to show that given $x,y \in \mathbb{C}^n$, the following holds $$ ||x + y||_{\infty} \leq ||x||_{\infty} + ||y||_{\infty} $$ Assume $x = a + ib, y = c + id, a,b,c,d \in \mathbb{R}$. $$ ...
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1answer
38 views

$x \in \mathbb R^d$, $B$(closed)$ \subseteq \mathbb R^d$. Where $\mathbb R^d$ is an inner product space. Proof help [duplicate]

$x \in \mathbb R^d$, $B$(closed)$ \subseteq \mathbb R^d$. Where $\mathbb R^d$ is an inner product space. Show that there exists a point $b_0 \in B$ such that $d(x,B) = \|x - b_0\|$ My attempt: ...
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61 views

Relation between Frobenius norm and eigenvalues

I'm considering a stochastic multivariate process, the stability of which implies that all eigenvalues $\lambda_i$, $i = \overline{1,n}$ of a certain square real-valued matrix $A$ lie within the ...