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

learn more… | top users | synonyms

3
votes
0answers
74 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 ...
3
votes
0answers
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 $|a| \neq 0$, ...
3
votes
0answers
107 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 ...
3
votes
0answers
240 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 ...
2
votes
3answers
412 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.
2
votes
2answers
311 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?
2
votes
3answers
116 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 ...
2
votes
1answer
519 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 ...
2
votes
3answers
1k views

Examples of metric spaces which are not normed linear spaces?

Give an example of a metric space which is not a normed linear space. Justify your example.
2
votes
3answers
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 $$ \sqrt{\sum_{i=1}^n |v_i|^2} \leq \sum_{i=1}^n |v_i|, $$ ...
2
votes
2answers
534 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= ...
2
votes
2answers
118 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!
2
votes
2answers
101 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 ...
2
votes
2answers
507 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 ...
2
votes
4answers
649 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 ...
2
votes
2answers
38 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 ...
2
votes
4answers
151 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 ...
2
votes
2answers
86 views

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 ...
2
votes
1answer
1k views

Why is the operator norm of a diagonal matrix it's largest value?

I read this in my textbook have tried working through it - I keep getting max 2-norm(Ax), which is just the magnitude of Ax. How should I do this proof? (note, this is not for homework, I'm just ...
2
votes
3answers
625 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 ...
2
votes
1answer
274 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 ...
2
votes
3answers
84 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. ...
2
votes
2answers
106 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
2
votes
2answers
126 views

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

How would you prove that $\|x\|_2 \geq \|x\|_1$, or in other words that $$\sqrt{\int_0^1|f(x)|^2 dx} \geq \int_0^1|f(x)|dx \quad?$$
2
votes
1answer
128 views

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 ...
2
votes
1answer
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 ...
2
votes
2answers
35 views

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 ...
2
votes
2answers
50 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 ...
2
votes
2answers
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. ...
2
votes
1answer
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 ...
2
votes
1answer
412 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 ...
2
votes
2answers
134 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 ...
2
votes
1answer
234 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}| ...
2
votes
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
votes
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
votes
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}}\ =\ ...
2
votes
2answers
228 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 ...
2
votes
3answers
152 views

Proving $|x|$ is a norm in $\mathbb {R}^n$

I need some direction on how to start on showing that $| x+y|\leq|x|+|y|$ in $\mathbb R^n$. Note that $$ |x|=\left(\sum\limits_{j=0}^n x_i^2\right)^{1/2} $$ Thank you, Klara
2
votes
1answer
315 views

The openness of the set of positive definite square matrices

Let $\mathbb{R}^{n\times n}$ be the vector space of square matrices with real entries. For each $A\in \mathbb{R}^{n\times n}$ we consider the norms given by: $$ \displaystyle\|A\|_1=\max_{1\leq j\leq ...
2
votes
2answers
311 views

Finding $L^2$ norm of solution of ODE

I have a linear differential equation with real constant coefficients $$ \sum\limits_{i=0}^3 a_i y^{(i)}(x)=0 $$ with initial conditions $y^{(i)}(0)=y_i\in\mathbb{R}$ where $i=0,1,2$. I need to find ...
2
votes
1answer
279 views

looking for a norm inequality

I want an inequality of the form : $\Vert a - b \Vert^2 \leq k.(\Vert a\Vert^2 + \Vert b\Vert^2)$ ? where k is a constant. The norm in consideration is the euclidean norm, and $a$ and $b$ are ...
2
votes
1answer
486 views

A question on linear transformation

Let $T$ be a linear transformation on $\mathbb{R}^{4}$ whose standard matrix is $$\left(\begin{array}{rrrr} 1 & -1 & -1 & -1\\ 1 & 1 & 1 & -1\\ 1 & 1 & -1 & 1\\ ...
2
votes
2answers
17 views

Norms — Distance Between Vectors

Which two of the vectors $u=(-2,2,1)^T$, $v=(1,4,1)^T$, and $w=(0,0,-1)^T$ are closets to each other in distance for (a) the Euclidean norm? (b) the infinity norm? (c) the 1 norm? I believe I know ...
2
votes
2answers
72 views

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 ...
2
votes
2answers
55 views

Condition for a norm be absolute

Let $\|x\|_B\mathrel{\mathop:}=\sqrt{x^{t}Bx}$, where $B \in \mathbb{R}^{n\times n}$ is a symmetric and positive semidefinite matrix. If $\mid x\mid = (|x_1|,|x_2|,\ldots,|x_n|)$, I want to show that ...
2
votes
2answers
60 views

Frobenius norm bound

Is there any way to bound Frobenius norm of a product of square matrices A,B and a vector x in the following way: $$ \|ABx\|≤ \|Ax\|\text{ and }\|B\| $$
2
votes
1answer
43 views

Submultiplicativity stronger than triangle inequality?

I would like to ask a question about matrix norm. Is the submiltiplicativity property always stronger than the triangle inequality? So, if i prove for a matrix norm that it's submultiplicative, i ...
2
votes
1answer
81 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 ...
2
votes
2answers
93 views

Parallelogram law in normed vectorspace withour an inner product.

Let $V$ be any $\mathbb{K}$-vectorspace with norm $\|\cdot\|$ I know that the Parallelogram law holds if the norm is induced by some inner product $\langle\cdot,\cdot\rangle$, i.e. $$ ...
2
votes
2answers
109 views

Must vectors in $\mathbb{R}^n$ have their “tail” at origin?

I was looking the definition for an $n$-sphere centered at origin with radius $r$: $$\mathbb{S}^n = \{v \in \mathbb{R}^{n+1} : ||v|| = r \}$$ Although I understand that the $||v|| = r$ condition ...