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 operator $g\mapsto \int fg$

Let $T_f:(C([a,b],\mathbb C), \lVert \cdot \rVert_1) \to \mathbb C$ with $g\mapsto \int_a^b f(x)g(x) dx$ for any given $f\in C([a,b],\mathbb C)$. I have to find the norm of $T_f$. I started with: ...
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431 views

Minimize distance between 2 functions

Just so you know, this is a homework question, and I basically need help with the steps to solve this problem. I understand what it's asking; however, my attempts haven't worked out, and I'm probably ...
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276 views

Norm with special conditions

Let $N$ be a norm on $\mathbb R^2$, such that $N ( \mathbb Z^2) \subset \mathbb N $, where $\mathbb Z^2 =\{ (a,b)\mid a\mbox{ and }b \mbox{ are integers}\}$. Help me to prove that for $u$, $v$ fixed ...
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62 views

Inequivalent norms (given by different inner products) on infinite dimensional Hilbert space.

I have this question in reviewing for my exam. Let $H$ be an infinite dimensional Hilbert space. Write down an inner product on $H$ that gives a norm inequivalent with the original norm. Is $H$ ...
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69 views

Do these limits commute?

Given a sequence of functions $f_{n,m}:\mathbb{R}^{n} \to \mathbb{R}$, suppose that $$\displaystyle lim_{m} f_{n,m}(x)$$ exists almost everywhere (for any fixed n) and also suppose that ...
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34 views

A General Question arises about norms and limits [closed]

Suppose $x_n \to x_0$, i.e. $\lim x_n=x_0$. How we can show that $||x_n|| \to ||x_0||$, that is $\lim_{n \to \infty} ||x_n||=||x_0||$, so $\lim_{n \to \infty} ||x_n|| = ||\lim_{x \to \infty} x_n||$?
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23 views

Cauchy Sequences Lemma in Vector Space E

I ran into a Lemma. Suppose $||.||_1$ and $||.||_2$ are two norms in vector spapce E, such that $||.||_1$ and $||.||_2$ are equivalent norms and {$x_n$} is an equivalent in E, then {$x_n$} is ...
3
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61 views

Are isometries always linear?

Let $E$ be a finite dimensional vector space (over a field of characteristic zero) and $f : E \rightarrow E$ be an isometry fixing 0. Must $f$ be linear in this case ? Note : I am NOT assuming that ...
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252 views

Equivalence of Frobenius norm and trace norm

According to [1], [2] and other related publications, the following holds for any matrix $X$: $$\| ...
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42 views

Norms on $\mathbb{Q}$

So with respect to the metric $d(x,y)=|x-y|$ induced by the standard absolute value, the real numbers can be constructed as a completion of $\mathbb{Q}$. With respect to the metric $d_p(x,y)=|x-y|_p$ ...
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139 views

Two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on $X$, and a function $f:X\to Y$ Fréchet differentiable with one of the norms but not with the other one?

There is a theorem that if $f : (X,\|\cdot\|_{X1}) \to (Y,\|\cdot\|_{Y1}) $ is Fréchet differentiable, then replacing the norms with some equivalent norms $\|\cdot\|_{X2}$ and $\|\cdot\|_{Y2}$ ...
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101 views

Proving norm equivalence in $W^{1-1/p,p}(\Omega)$

Define for $p\in [1,\infty)$ and $\Omega=(0,1)^N\subset\mathbb{R}^N$ $$W^{1-1/p,p}(\Omega)=\left\{u\in L^p(\Omega): \ \int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy<\infty\right\}$$ ...
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196 views

Prove this equality in a Hilbert Space $H$ with Riesz's Representation Theorem.

For $x \in H$, prove that $\sup_{\|z\| = 1} \langle x , z \rangle= \| x \|$ Here is a quick one. If someone could improve this it would be great Proof By Cauchy Schwarz, $\langle x,z \rangle ...
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178 views

The norm of an operator

Let $\rho(x)$ be a weight function in a unit sphere, such that \begin{equation} \begin{array}{l} \displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\ \displaystyle 2. \rho(x)\in ...
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76 views

Exercise from textbook about norm

The below diagram included a relevant page and the question 4 of the exercise which i am not sure how to do. Any hint or solution are welcome Attempts: a) I have done it b) i have tried to show that ...
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92 views

Is $\rho(f) := \mu(|f| \geq \epsilon), \epsilon >0$ some kind of generalized norm?

Let $\rho(f) := \mu(|f| \geq \epsilon)$ where $\epsilon >0$ is a constant and $\mu$ is a measure and $f$ is a measurable function. Then $f\equiv 0 \rightarrow \rho(f) = 0$ when $|a| \neq 0$, ...
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109 views

Confusion about matrix norms

Reading the wiki article I get confused about matrix norms. My question, is it true that $$\lVert Mx \rVert \leq C(\sup_{ij}{M_{ij}}) \lVert x \rVert$$ where $M$ is a matrix and $x$ is a vector and ...
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246 views

Minimizing maximum absolute column sum norm of the residual between a matrix and its $k$-rank approximation

Let $X \in \mathbb{R}^{m\times n}$ be a matrix with rank $r$. How can we find the optimal $\tilde{X} \in \mathbb{R}^{m\times n}$ whose rank is $k$ where $k\leq r$ and the reconstruction error in ...
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429 views

How to prove $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$?

I'm trying to show that $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$. A hint would be nice.
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2answers
318 views

What is the difference between $L^2$ norm and $\ell^2$ norm?

I can find no precise definitions on the internet for the $L^2$ and $\ell^2$ norms. Certain websites keep switching between the two. Can someone please help me?
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117 views

Convergence of a pair linearly independent elements of a vector space

Let $v_1$ and $v_2$ be linearly independent elements of some normed vector space $V$. We have a sequence $(a_n,b_n)$ and are told that $a_nv_1+b_nv_2$ converges to the zero element. Is it the case ...
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556 views

Why do we need semi-norms on Sobolev-spaces?

I have been studying Sobolev spaces and easy PDEs on those spaces for a while now and keep wondering about the norms on these spaces. We obviously have the usual norm $\|\cdot\|_{W^{k,p}}$, but some ...
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126 views

Why is $\sqrt{\sum_{i=1}^n |v_i|^2} \leq \sum_{i=1}^n |v_i|$ true?

Sorry if this is very basic but here's a question. Let $\mathbf{v}=(v_1,\ldots, v_n)\in k^n$ where $k=\bar{k}$. Why do we have $$ \sqrt{\sum_{i=1}^n |v_i|^2} \leq \sum_{i=1}^n |v_i|, $$ ...
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557 views

How do you prove the $p$-norm is not a norm in $\mathbb R^n$ when $0<p<1$?

I see that it fails to satisfy the triangle inequality by example but I don't see how to prove this is the case for all $0 < p < 1$. The definition I am using for $p$-norm is $$ \|A\|_p= ...
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45 views

Determination of some operator norms

I have to determine the operator norms, the kernels and the images of the following 2 maps: 1) $F_1 :\{x\in C^0([0,10],\mathbb R)|x(0)=0\}\rightarrow C^0([0,10],\mathbb R)$ ...
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123 views

How does one prove $\left\lVert\sum\limits_{n=1}^{\infty}f_n\right\rVert\le \sum\limits_{n=1}^{\infty}\lVert f_n\rVert$?

Can the triangle inequality for norms works for infinite some as well? Hope that someone can give some hints to prove it or give a counterexample. Thanks!
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108 views

What is the definition of the norm

Let $x$ and $y$ be in $ \mathbb{R}^{n}$. I know from the definition of norm that $\|x\|=\sqrt{\sum_{1}^{n}x_{i}^{2}}$. Can anyone tell me what will be the norm of $\|x-y\|$? Is it ...
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547 views

$q$-norm $\leq$ $p$-norm [duplicate]

[Ciarlet 1.4-8] If $0 < p < q$, show that $$\left(\sum_{i=1}^n|v_i|^q\right)^{1/q}\ \leq\ \left(\sum_{i=1}^n|v_i|^p\right)^{1/p}$$ Somebody knows how prove that? Thanks in adavance for the ...
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708 views

Prove that $N(\gamma) = 1$ if, and only if, $\gamma$ is a unit in the ring $\mathbb{Z}[\sqrt{n}]$

Prove that $N(\gamma) = 1$ if, and only if, $\gamma$ is a unit in the ring $\mathbb{Z}[\sqrt{n}]$ Where $N$ is the norm function that maps $\gamma = a+b\sqrt{n} \mapsto \left | a^2-nb^2 \right |$ I ...
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41 views

Is $u^TAu \geq 0$ true for all symmetric matrices $A$?

we know from the definition of inner product and norm, that $u^Tu$ is always larger than zero, except the case where $u=0$ at which case it is zero. I came across a question that infers that $u^TAu ...
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4answers
156 views

Prove that $||x|-|y|| \leq |x-y|$ [duplicate]

$||x|-|y|| \leq |x-y|$ when $(x,y \in R^k)$ In Principles of MA(Rudin), the author said one sees easily that $||x|-|y|| \leq |x-y|$ when $(x,y \in R^k)$ (p.88, Rudin) from the triangle ...
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Generalizing norms for modules

An normed space is defined as a vector space V plus a norm operation over $V$. Is it meaningful to generalize this notion to modules, where one is dealing with rings instead of fields? What I'm ...
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691 views

norm of a linear operator

On http://www.proofwiki.org/wiki/Definition:Norm_(Linear_Transformation) , it is stated that $||Ah|| \leq ||A||||h||$ where $A$ is an operator. Is this a theorem of some sort? If so, how can it be ...
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283 views

Matrix norms involving singular values

The induced 2-norm of a $m \times n$ matrix $A$ is known as $$ ||A||_2 = \underset{||x||=1}{\text{max}} ||Ax||_2 = \sigma_1,$$ where $\sigma_1$ is the largest singular value of $A$. Then, is it ...
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85 views

Does a norm have to map to $\mathbb R$?

Wikipedia defines a norm on a vector space $V$ as a function $p : V \mapsto \mathbb R$. I've seen this defined similarly elsewhere. However, it seems to me that a real codomain isn't always necessary. ...
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109 views

is product of norms convex?

Is a function of the form $f(x) = \|x\|_1\|x\|_2$ convex in x? I have tried plotting it in wolfram alpha and it appears convex, althought I ahve not been able to show it yet
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126 views

Proving that $\|x\|_2 \geq \|x\|_1$?

How would you prove that $\|x\|_2 \geq \|x\|_1$, or in other words that $$\sqrt{\int_0^1|f(x)|^2 dx} \geq \int_0^1|f(x)|dx \quad?$$
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Convergence of $\frac1m(I+A+A^2+\cdots+A^{m-1})$

Let $A$ be an $n\times n$ matrix of nonnegative entries such that $A_{i1}+A_{i2}+\cdots+A_{in}=1$ for all $i\in\{1,2,\ldots,n\}$. What does $A$ have to satisfy so that the sequence ...
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20 views

Operator in $\mathbb R^2$

I am a bit confused, can someone help me with the following? Is there an operator $T$ in $\mathbb{R}^2$ such that: $\parallel u \parallel +\parallel v\parallel = \parallel T(u+v)\parallel$ for every ...
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Spectral Radius and Norm of multiplied vector

Let $\mathbf{A}$, $\mathbf{B}$ be square matrices of equal dimensions, $\mathbf{w}$ a vector of compatible dimensions and $\rho$ be the spectral radius operator. Does the following hold? If $\rho ...
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51 views

Matrix-Vector Product

Suppose I have the expression $\lVert\mathbf B \cdot\hat n\lVert=1$, where $\mathbf B$ is a matrix and $\hat n$ is a unit vector (both can have any dimensions, as long as they are compatible). What ...
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80 views

Can anyone explain this isometry to me? $T: (\mathbb R^n,\|\cdot\|_1)^*\to (\mathbb R^n,\|\cdot\|_{\infty})$, $ T(x)(y) = \sum_{i=1}^n x_i y_i$

Can anyone explain this isometry to me? $$T: (\mathbb R^n,\|\cdot\|_1)^*\to (\mathbb R^n,\|\cdot\|_{\infty}),\qquad T(x)(y) = \sum_{i=1}^n x_i y_i$$ I don't get what the domain and image of $T$ are. ...
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49 views

Calculate $\|f\|$ with $f(x) = \int_{-1}^{1}x(t)dt - 2x(0)$

In $C[-1,1]$, consider the norm $\|x\| = \max\{x(t)\mid t \in [-1,1]\}$ and $f(x) = \int_{-1}^{1}x(t)dt - 2x(0)$, $x \in C[-1,1]$. Find $\|f\|$ Hi everybody. I got stuck in this problem. I can ...
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500 views

Why is the matrix norm $||A||_1$ maximum absolute column sum of the matrix.

By definition, we have: $||V||_p=\sqrt[p]{\displaystyle \sum_{i=1}^{n}|v_i|^p}$ $||A||_p=sup\frac{||Ax||_p}{||x||_p}$, $x\not=0$ and if $A$ is finite, we change sup to max. However I don't really ...
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137 views

Show that $D =\{ x + y \mid x \in (0,1) ,y \in [1,2) \}$ is open or closed

I have $\mathbb{R}$ with the euclidian metric $|x-y|$ for $x,y\in \mathbb{R}$. I want to show with arguments that the set $D =\{ x + y \mid x \in (0,1) ,y \in [1,2) \}$ is open or closed. As a ...
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239 views

Property for Norms of Matrices

I am having trouble with the following problem: Show that the vector norm $||x||_1$ gives the subordinate matrix norm: \begin{equation} ||A||_1 = \max_{1\leq j\leq n}\sum_{i=1}^n|a_{ij}| ...
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1answer
96 views

Derive $\frac1n \|x\|_p^p \leq \|x\| \leq n^{p/2}\|x\|_p^p$ from Holder's inequality?

Given a vector $x = (x_1, \dotsc, x_n)\in \mathbb{C}^n$, I wanted to compare $|x_1|^p + \dotsb + |x_n|^p$ to $\|x\|^p$. I discovered that if $m=\max_i|x_i|$, we have $$m^p \leq \|x\|^p \leq ...
2
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1answer
2k views

How to prove triangle inequality for $p$-norm?

Well, I've been studying metric spaces and to make the cartesian product of metric spaces a metric space I've heard of the $p$-norm defined in $\mathbb{R}^n$. So if $\mathcal{M}=\{M_i : i\in I_n\}$ is ...
2
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1answer
70 views

Simple doubt about dual norm

If $(X, \|\cdot\|)$ is a normed vector space, then $$\|F\|_{X^{\prime}}\ =\ \sup_{x\in X-\{0\}}\frac{|F(x)|}{\|x\|},$$ by definition. Then I want prove that, $$\|F\|_{X^{\prime}}\ =\ ...
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254 views

Derive steepest descent vector subject to norm constraint

I am currently working through an old textbook Practical Optimization by Gill, Murray and Wright (c 1982) who make some derivations which seem correct, but I am unable to duplicate. In the equations ...