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

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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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2answers
70 views

norm of inverse less than 1

I just wanna ask if what I am doing here make sense: Let $\epsilon$ be arbitrary positive number. Choosing $\epsilon$ and let it approaches 0, I would like to have $||(I-\epsilon A)^{-1}|| < 1$. ...
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71 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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2answers
69 views

Linear functional $\mathscr{L}(E,F)$

Let $\mathscr{L}(E,F)$ denote the space of all linear functionals from $E \to F$. Let $\mathscr{C}(E,F)$ denote the space of continuous linear functionals from $E \to F$. My question: How to prove ...
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2answers
511 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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1answer
199 views

Matrix Norm set #2

As a complement of the question Matrix Norm set and in order to complete the Problem 1.4-5 from the book: Numerical Linear Algebra and Optimisaton by Ciarlet. I have this additional conditions: (3) ...
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1answer
87 views

Study the equivalence of these norms

I have two Hilbert spaces $H_1$ and $H_2$ and I consider a set of functions $f$ which decompose as $f=g+h$ with $g\in H_1$ and $h\in H_2$. I know that this decomposition is unique. So I define the ...
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81 views

Matrix norm applications?

There'are many different ways to calculate Matrix norm. But once calculated, what is the practical use/application of it (e.g. in computer programming)? Or does it let define something that can be ...
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1answer
68 views

What is $L^1$ norm for $n$-fold functional space of integrable function?

Consider a space of $n$-fold real-valued integrable functions, $X \doteq [L^1([0,T])]^n$. Some use an alternative notation like $X \doteq L^1([0,T]; \mathbb{R}^n)$. Does $f\in X$ mean ...
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1answer
128 views

On norm selection for the solution of an overdetermined linear system

I am considering the following linear system: $Ax = b$ Where: $A$ is $9000 \times 139$ $x$ is $139 \times 1$ and sparse $b$ is $9000 \times 1$ Most of the resources I have found online point to ...
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87 views

All matrix/vector norms induce the same topology?

From Wikipedia all norms on $K^{m \times n}$ are equivalent; they induce the same topology on $K^{m \times n}$. This is true because the vector space $K^{m \times n}$ has the finite dimension $m ...
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1answer
384 views

Calculating the Norm of an operator in $L^2(0,1)$

If I have the following operator for $H=L^2(0,1)$: $$Tf(s)=\int_0^1 (5s^2t^2+2)(f(t))dt$$ and I wish to calculate $||T||$, how do I go about doing this: I know that in $L^2(0,1)$ we have that ...
3
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1answer
41 views

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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80 views

Are these two definitions for dual norm equivalent

Suppose there is a p norm, also is a function $\left\|x\right\|_p$, the dual function at $z$ of it is, $$\sup\limits_x\left\{ z^{T} x-\left\|x\right\|_p\right\}$$ The second is, $$ \sup\limits_x ...
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2answers
146 views

Are the norms $\|\cdot\|_1$ and $\|\cdot\|$ equivalent?

Let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $\ell^1$, suppose $x=\{x_n\}\in\ell^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$. Are the norms $\|\cdot\|_1$ and $\|\cdot\|$ equivalent?
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1answer
117 views

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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1answer
299 views

Matrix Norm set

I need help with this problem: Let $\|\cdot\|$ and $\|\cdot\|^{\prime}$ two matrix norms, and consider the relation $$\|\cdot\| \leq \|\cdot\|^{\prime}\ \Leftrightarrow\ \|A\| \leq \|A\|^{\prime},$$ ...
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0answers
60 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
91 views

Two inequalities related to norm

We have some difficulties in the following problem: Let $H$ be a real Hilbert space. Find $\alpha>0$ such that $$ \langle\frac{u}{\sqrt{\|u\|}}-\frac{v}{\sqrt{\|v\|}}, u-v\rangle\geq ...
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1answer
213 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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167 views

Effect of doubly stochastic matrix on vector norm

Let $D$ be a $N \times N$ doubly stochastic matrix, $x$ be a $N$ dimensional vector. What is the relation between $\Vert Dx \Vert_2$ and $\Vert x \Vert_2$? In addition if $\Vert x \Vert_2=1$, what ...
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1answer
45 views

Order of infinite dimension norms

I know that $$\|{f}\|_{L^1(0,L)}\leq\|{f}\|_{L^2(0,L)}\leq\|{f}\|_{\mathscr{C}^1(0,L)}\leq\|{f}\|_{\mathscr{C}^2(0,L)}\leq\|{f}\|_{\mathscr{C}^{\infty}(0,L)}$$ But I don't know where to put in this ...
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43 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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1answer
69 views

Determining a norm from a quadratic form

If $B$ is a quadratic form over some space $V$, what is the norm determined by $B$? Is this the inner product $\langle Bu,Bv\rangle$? If not, and it is not possible to determine a norm from knowing ...
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1answer
453 views

Using Lax Milgram to find a weak solution in an intersection of Sobolev spaces

I am trying to prove the existence of a weak solution of the problem: $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ on the bounded open set $U\subset\mathbb{R}^n$ ...
3
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2answers
110 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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1answer
67 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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1answer
48 views

Regularity and the Varitational Inequality

Let $K = \left\{ v \in H_0^1(\Omega) \, : \, v \geq 0 \right\}$, further suppose $\Omega$ has the regularity property that $||v||_{H^2} \leq C(\Omega)||\Delta v||_{L^2}$, for all $v \in ...
3
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1answer
47 views

bound on $l_2$ error in approximating a vector with its $t$-sparse representation

How do I prove that for any vector $y\in \mathbb{R}^n$, and any positive integer $t$, \begin{equation} ||y-y_t||_2\:\leq\: \frac{1}{2\sqrt{t}}||y||_1 \end{equation} where $y_t\in\mathbb{R}^n$ is the ...
2
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1answer
380 views

Relation between L2 norm and L1 norm of two vectors

For two given $N\times 1$ vectors $x,y$ , if $||x||_2\geq ||y||_2$, can I say that $||x||_1\geq ||y||_1$? If not give an example. Converse of this question has been asked here.
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44 views

Finding an orthornormal basis given a bilinear form

Take $ \langle \cdot,\cdot\rangle$ to be a nondegenerate symmetric bilinear form (which may not be positive definite). Take $\Vert x\Vert = \vert \langle x,x\rangle \vert^{1/2}$. I want to show that ...
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4answers
531 views

Normalizing a matrix

I came across a step in an numerical algebra algorithm that says "Normalize the rows of matrix A such that they are unit-norm. Call U the normalized matrix." I do something like this: ...
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2answers
30 views

Clarification with a identity

I have the following inequality which comes in proof of triangle inequality. $\|a+b\|^2=\|a\|^2$+$\|b\|^2+2\Re\langle a|b\rangle\le\|a\|^2+\|b\|^2+2|\langle a|b\rangle |$ I don't know where the ...
2
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1answer
100 views

Operator norm estimate

Let $H$ be a Hilbert space with orthonormal basis $(e_{j})_{j\in\mathbb{N}}$. Furthermore, let $B\colon H\rightarrow C[a,b]$ be a bounded operator. According to the Riesz-Frechet theorem there is ...
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109 views

What are norms used for?

These two questions are quite similar to this one, so I apologise if this irritates anyone. Also, I suspect that a lot can be said in the answer, so I am really just looking for some main points ...
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2answers
187 views

Link between the norm $1$ of a matrix and its biggest eigenvalue

I am working on a set of matrices for a project, studying their highest eigenvalue, let's call it $\lambda_{1}$. I was curious and plotted the norm 1 of the matrix, ie $ \frac{1}{n^{2}}\sum_{i,j} ...
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45 views

Showing a function is a norm.

I'm trying to prove that $\Vert v\Vert :=\langle v,v\rangle^{1/2} $ defines a norm, but I'm having trouble with the triangle inequality. $\Vert u+v\Vert=\langle u+v,u+v\rangle^{1/2}=(\langle u,u ...
2
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3answers
307 views

Norm equivalence of a vector norm and its induced matrix norm using compactness argument

I have a theorem in my book on matrix computations that states the following: A vector norm and its induced matrix norm satisfy the inequality: $\|Ax\|\leq \|A\|$$\|x\|$ where A $\in R^{nxn}$ and x ...
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95 views

How to show for a PSD matrix $A$ that $\left \| \left ( A+I \right )^{-1} \right \| \leq \frac{1}{1+\sigma _{\min}\left ( A \right )}$?

If $A \in \mathbb{C}^{n \times n}$ is positive semidefinite, show that $\left \| \left ( A+I \right )^{-1} \right \| \leq \frac{1}{1+\sigma _{\min}\left ( A \right )}$, where $\sigma _{\min}\left ( A ...
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1answer
495 views

Norm of the linear functional

Could you help me, please with the following question? There is a linear functional $A : C_{[0;1]} \rightarrow \mathbb{R}$, such that $$ Ax=\int_{a}^{b}x(t)\varphi(t)dt $$ where $\varphi$ is a fixed ...
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2answers
60 views

On Real Hamilton Ring ..

i know the definition of real hamilton ring but if we said ,$ I$ is the ring of integral hamilton what does this mean ? what is the properites that word , integral , adds to the structure of ...
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120 views

Norm of a function, Smoothness Penalization

I am seeking for some intuition why norm (for any reasonable norm on functions) of a function is smaller if the function is smoother.
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3answers
632 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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91 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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55 views

Boundary value problem domain, norm?

I'm starting some research on a boundary value problem for a memoire for the 2nd semester of a masters, and my mentoring professor asked me to find the answer to the following question: For $u \in ...
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1answer
13 views

Positivity of a map in $(l^\infty(X))^*$

Let $X$ be a set and $\varphi: l^\infty(X)\to\mathbb{R}$ be a linear map such that $||\varphi||=1$ $\varphi(1_X)=1$ I am trying to prove that $\varphi(f)\ge 0$ for all $f\ge 0$, but all my ...
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1answer
33 views

Matrixnorm, “max” function

According to my lecture notes, the following equality holds: $ \frac{1}{\sqrt{n}}$ $\max_{x \neq 0}$ $\frac{\|Ax\|_{2}}{\frac{1}{\sqrt{n}}\|x\|_{2}}$ = $\max_{x \neq 0}$ ...
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2answers
61 views

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
353 views

Submultiplicative Matrix Norm

Given $I$ is the $n\times n$ identity matrix, $F$ an $n\times n$ matrix with $\|F\|_{M} < 1$ for some submultiplicative norm $\|.\|_M$, by which formula is $(I+F)^{-1}$ calculable? And is it ...
2
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1answer
110 views

finding operator norm $T_N$

How do find the operator norm of , $T_N\colon c_0\to \Bbb R$ given by $T_N(y):=\sum_{j=1}^Nx_jy_j$ ,when $N$ is a integer . TIA