Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [norm]

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

1
vote
1answer
12 views

Is there a special relationship between a norm on a vector space V, and the operator norm $ \mathcal{L}(V, \mathbb{R)}$?

Let $T$ be a linear operator in $\mathcal{L}(V)$. An operator norm is denoted as $||T||$, where it is the smallest $M$, such that $||T(v)||$ $\le$ $M||v||$ for any $v \in V$. A norm on the vector ...
0
votes
1answer
17 views

Norm of a bounded linear functional.

Let $X=(\mathbb R^2, \|.\|_3)$ be a real normed space, where $\|(x_1,x_2)\|_3=[|x_1|^3+|x_2|^3]^{1/3}$. How to find the norm of bounded linear functional $ax+by$? I tried this way: $|ax+by|\leq |a||x|...
2
votes
1answer
38 views

Show that $\sum_{k=0}^{N} |P(k)| \leq C(N) \int_{0} ^{1} |P(t)| dt $.

I’m attending a functional analysis course and I am given to solve this problem as an exercise but I’m a little bit disoriented and I don’t know what tools I can use to get it. Show that, for each $...
1
vote
0answers
17 views

Can we find a natural norm smaller than the infinite norm for this special matrix?

Let $\hat{\bf H}$ be a $p\hat{N}\times p \hat{N}$ sparse matrix consisting of $p\times p$ blocks, where each block is of size $\hat{N}\times\hat{N}$. The values in $\hat{\bf H}$ is illustrated below (...
0
votes
0answers
11 views

References of weighted norm distance relaxation?

In my practical work, I need to measure the distance between two vector $a$ and $b$ with dimensions weighted by vector $w$ in $\mathbb{R}^n$. I need to learn the weight for the distance, then use the ...
0
votes
0answers
8 views

Find norm of image convolution

Suppose there is matrix A that is a black and white image. There's also matrix B - a filter. ...
0
votes
0answers
25 views

Constraint Least Square (CLS) optimization with both equality and inequality constraints

Given is the following CLS problem: \begin{align} (1)&&\min_{\mathbf{x} \in \mathbb{C}^n} &\|A \mathbf{x} - \mathbf{b}\|,\\ \text{ subject to:}&&&\\ \text{(C1)}&&\...
0
votes
1answer
20 views

Showing $f:=f_{1},…,f_{n}\in L^{p}(\mu)$ for $f_{i} \in L^{p_{i}}$

Let $f_{1},...,f_{n}:X \to \bar{\mathbb R}$ measurable, while $f:=f_{1}\times...\times f_{n}$, and $p_{1},...,p_{n} \in [1,\infty]$ where $f_{i}\in L^{p_{i}}(\mu), \forall i\in \{1,...,n\}$. Note $\...
-1
votes
0answers
11 views

Maximum of matrix from eigenvalues

Can anyone help in solving following question: Find the maximum of ((x1 + 4x2)^2)/(x1^2 + x2^2). For what matrix A is this (||Ax||^2)/(||x||)^2. So (||Ax||^2) can be written as x.TA.TAx. From our ...
0
votes
0answers
14 views

Question about smoothness of a Banach space.

Define $\delta: [0,2] \to [0,1]$ defined by $$\delta_U(\epsilon) = \inf \bigg\{\frac{1}{2}\bigg(2-\|u_1+u_2\|\bigg): \ u_1, u_2 \in U^0, \|u_1-u_2\| \geq \epsilon\bigg\},$$ where $U^0$ is the ...
3
votes
0answers
25 views

Find a condition such that the spectral radius for a special matrix is smaller than 1 (or a matrix norm smaller than 1).

We need a help to find a reasonable condition such that the spectral radius for a special matrix $\mathbf{J} \otimes\hat{\mathbf{G}}\hat{\mathbf{W}} + \mathbf{I}\otimes\mathbf{\hat{H}}$ is smaller ...
0
votes
1answer
21 views

Converting a first norm into a linear program

I want to convert the following problem into a linear program. $$ \min_{x \in \mathbb{R}^n} \qquad \lvert\lvert Ax - b \rvert\rvert_1$$ Here $A \in \mathbb{R}^{m\times n}$, $x \in \mathbb{R}^n$ and $...
0
votes
0answers
14 views

Norm of least squares residual

Let $A=QR$ be the full $QR$ factorization of $A\in\mathbb{C}^{m\times n}$, $m\ge n$, where $$Q=\begin{bmatrix} \hat{Q}_1 & \hat{Q}_2 \end{bmatrix}, R=\begin{bmatrix} \hat{R}_1\\ 0 \end{bmatrix},$...
-1
votes
0answers
29 views

Show that $\Vert x\Vert_* := \Vert Ux\Vert $ is a vector norm [closed]

Let $\Vert * \Vert$ be a Vectornorm in $\mathbb{R}^n$ or $\mathbb{C}^n$ and $U \in \mathbb{R}^{n\times n}$ a regular matrix. How can I show that $\Vert x\Vert_* := \Vert Ux\Vert $ is also a ...
0
votes
1answer
28 views

Evaluating the value of the norm of a function

I am trying to show: $$\|f_\alpha \| = \| \alpha \|_{\ell^\infty} $$ Given that: $$f_\alpha:\mathbb{R}^2 \rightarrow \mathbb{R}, \quad f_\alpha = \alpha_1 x_1 + \alpha_2 x_2$$ ($\mathbb{R^2}$ is ...
1
vote
0answers
47 views

Find $\|f\|^2$ in two ways

Question: Consider the Hilbert space $L^2(\mathbb R/\mathbb Z)$ of periodic measurable functions. Recall that it has a Hilbert space basis consisting of functions $\chi_n(x)=e^{2\pi i n x}$ where ...
1
vote
1answer
48 views

derivative of the Euclidean norm of matrix and matrix product

I have two matrices $A = \left[ {\begin{array}{*{20}{c}} 3&7&9&1\\ 4&1&2&3\\ 5&6&3&7\\ 2&4&3&7 \end{array}} \right]$ and $B = \left[ {\begin{array}{*{...
0
votes
2answers
35 views

Dual norm of $l_1$ of is $l_\infty$

I am trying to show that $l_1$ norm's dual norm is $l_{\infty}$ norm. I have proceeded like the following: $||z||_D = \sup \{z^Tx| ||x||_1\leq 1 \}$ Then: $ z^Tx \leq \sum_{i=1}^n |z_ix_i| = \sum_{i=...
2
votes
1answer
52 views

norm of difference of two matrices.

For any two symmetric $n\times n$ matrices $A$ and $B$, let their eigenvalues be ordered from largest to smallest. How to prove that for eigenvalues $$|\lambda_k^A-\lambda_k^B| \leq \|A-B\| \ \text{...
4
votes
1answer
52 views

Show there is an $n$-th degree polynomial $p(x)$ such that $||f(x)-p(x)||_\infty \leq ||f(x)-q(x)||_\infty$.

Show that for each $f \in C[0,1]$ there is an $n$-th degree polynomial $p(x)$ on $[0,1]$ such that $||f(x)-p(x)||_\infty \leq ||f(x)-q(x)||_\infty$ for any other $n$-th degree polynomial $q(x)$. This ...
6
votes
3answers
65 views

Prove that $\Vert\cdot \Vert^2:X\to \Bbb{R},$ where $X$ is a vector space, is convex

Let $X$ be a vector space. I was able to prove that $\Vert\cdot \Vert:X\to \Bbb{R},$ is a convex function, i.e., for all $x,y\in X$ and $\lambda \in [0,1],$ \begin{align} \Vert \lambda x+(1-\lambda)y ...
1
vote
2answers
33 views

Point $x \in \mathbb{R}^n$ that minimizes sum of distance squares $\sum_{\mathcal{l}=1}^{k} \Vert x-a^{\mathcal{(l)}} \Vert _2^2$

Let $a^{(1)},...,a^{(k)} \in \mathbb{R}^n$. How can one find the point $x \in \mathbb{R}^n$, which minimizes the sum of distance squares $\sum_{\mathcal{l}=1}^{k} \Vert x-a^{\mathcal{(l)}} \Vert _2^2$...
0
votes
3answers
34 views

Finding norm of orthonormal basis?

I'm sorry i'm new here. I uploaded a pictures in order to make things simpler. I have three linearly independent vectors: v1= (1,1,0,0) v2=(1,-1,0,0) and v4=(0,2,0,0). As you may see from the ...
0
votes
1answer
42 views

How can a matrix act on a set

Question from a PhD entrance exam If $A=$ \begin{bmatrix} 2&-1\\-1&2 \end{bmatrix} and $X=\{x\in \Bbb R^2:\|x\|<1\}$ where $\|x\|=|x_1|+|x_2|$ Find $AX$. Now I know that how $\...
0
votes
0answers
18 views

Upperbounding the expected value of an L2-norm difference

I am trying to find an upperbound for $$E_I\left[\sum_{k=I}^n a_k^2 - \sum_{k=I}^n b_k^2\right],$$ where the expectation is taken over the random variable $I$. The tuples $a$ and $b$ are real and $I$ ...
0
votes
1answer
22 views

Equivalence in $\infty$-norm

If $\overline{M}:=\inf\left\{M:\mu(\left\{x:|f(x)|>M\right\})=0\right\}$ and $\overline{a}=\sup\left\{a:\mu(\left\{x:|f(x)|>a\right\})>0\right\}$ I want proves $\overline{a}=\overline{M}$ ...
-1
votes
1answer
27 views

Norm, gradient, vector

Can I say that the norm of a component of the vector is smaller than the norm of the whole vector? (in this case the gradient) $$\|v_x\| < \|\operatorname{grad} v\|$$
1
vote
1answer
28 views

L2-norm with estimated weights

Suppose I'm performing linear regression. My lecturer said the formula below can be used for estimating the weight vector that is passed to the L2-norm part of the loss function but he didn't ...
0
votes
0answers
12 views

induced norm of a one to one function

Let $V \subset C^0(\mathbb{R}, \mathbb{R})$ be a vector space and $ f: V \to \mathbb{R}^n : g \mapsto (g(i))_{i=1,\dotsc,n}$ a one to one function. Then in my lesson, they are talking about the norm ...
4
votes
1answer
61 views

For $p\in(1,\infty)$, why does $||f||_p=||g||_p=\left|\left|\frac{f+g}{2}\right|\right|_p$ imply that $f=g$?

Problem Statement. I am working on the following exercise: Prove that if $\mu$ is a measure, $p\in(1,\infty)$ and $f,g\in L^p(\mu)$ are such that $$||f||_p=||g||_p=\left|\left|\frac{f+g}{2}\...
0
votes
2answers
22 views

How to show that the following norms are equivalent?

Let $f \in C^1[0, 1]$. Then show that the norms $$|f| =\int_0^1|f(t)|\, dt+\max_{t \in [0, 1]}|f'(t)|$$ and $$\|f\|= \max_{t \in [0, 1]}|f(t)|+\max_{t \in [0, 1]}|f'(t)|$$ are equivalent. I have ...
1
vote
0answers
26 views

$L^2$ norm of a PDE

Iam working on the following problem: Let $\Omega \subseteq \mathbb{R}^n$ and $v$ be the solution of $(-\Delta+q-\lambda)v=F$ on $\Omega$ such that $q$ is bounded and $v=exp(-ax)u$ where $u\in H_{0}^{...
-1
votes
0answers
30 views

regularity of elliptic PDE

Iam studying the regularity of elliptic PDE, and Iam reading this pdf and I found it useful http://web.math.ucsb.edu/~grigoryan/246B/lecs/246B_ch5.pdf My question is how the bound of the inner ...
0
votes
0answers
23 views

Calculate the operator norm of the difference of two operators

Let $x_1,...,x_n \in [0,1]$, $\lambda_1,...,\lambda_n \in \mathbb{C}$ and let $S,T : C([0,1]) \to C([0,1])$ be the operators defined by $T(g)= \int_0^1 g(s)\, ds$ and $S(g) = \sum_{i=1}^n \lambda_i ...
1
vote
1answer
34 views

Question concerning an application of Cauchy-Schwarz

Specifically, the question is as follows: Prove that for every integrable real-valued $f:\mathbb{R}\rightarrow\mathbb{R}$, $$\left(\int_1^ef(x)dx\right)^2\leq\int_1^ex(f(x))^2dx.$$ I'm really ...
0
votes
0answers
14 views

Is Frobenius norm constant?

I came across an article that makes me doubt what I have learnt so far. For example, let us decompose matrix A into L and U (LU factorization) $A=\left( \begin{array}{cccc} 5 & 4 & 1 &...
0
votes
0answers
35 views

calculating norm with cut off function

Let $\chi$ be a smooth cut-off function, defined on an interval $[0,\infty)$ and it is equal 1 on $[0,X-1)$ and goes to 0 on $[X-1,X)$ then equal 0 on $[X,\infty)$.. Let $v=exp(-ax)$ on $[0,\infty)$.. ...
4
votes
0answers
38 views

Show the Polar Factor is the Closest Unitary Matrix Using the Spectral Norm

For a square matrix $A \in \mathbb{C}^{n \times n}$ with the singular value decomposition $A = U\Sigma V^*$, I want to show that $$\|A - P \|_{2} \leq \|A -W \|_{2}$$ Where $P = UV^{*}$ and $W$ is ...
4
votes
0answers
75 views

Variance of the Euclidean norm under finite moment assumptions

Let $X = (X_1,X_2 \cdots X_n)$ be random vector in $R^n$ with independent coordinate $X_i$ that satisfy $E[X_i^2]=1$ and $E[X_i^4] \leq K^4$. Then show that $$\operatorname{Var}(\| X\|_2) \leq CK^4$$...
0
votes
0answers
16 views

Discrete norm approximation of the $L^p$ norm for spline functions

In Theorem 5.2 in Lynche (1988) "A data reduction strategy for splines with applications to the approximation of functions and data", a bound for the difference between the $(l_2,t)$ and $L^2$ norms ...
-1
votes
0answers
26 views

$\|x-y\|_{L^2}^2\geq \|x\|_{L^2}^2-\|y\|_{L^2}^2$?

Is this true? \begin{equation} \|x-y\|_{L^2(\Omega)}^2\geq \|x\|_{L^2(\Omega)}^2-\|y\|_{L^2(\Omega)}^2 \end{equation}
0
votes
0answers
28 views

Lower bound for $\|x-y\|$

For $x$,$y$ in Hilbert space $\mathcal{H}$ I want a lower bound for \begin{equation} \|x-y\|_{\mathcal{H}}^2 \end{equation} I know \begin{equation} |\ \|x\|_{\mathcal{H}}-\|y\|_{\mathcal{H}}\ |\leq\|...
3
votes
1answer
78 views

When does $\Vert AB \Vert = \Vert A \Vert \Vert B \Vert$?

Motivation If $a$ and $b$ are vector, then thinking simply vector 2 norm, $\Vert a \cdot b\Vert = \Vert b\Vert \Vert a\Vert \cos(a,b) $, we know the difference is simply a ratio between the angle of $...
1
vote
1answer
21 views

Proof $\left\lVert g(x) \right\rVert_2 \leq \ln(1+\left\lVert x \right\rVert_2^2) \text{ for all x}\in \mathbb{R}^n $ differentiable

Let $g:\mathbb{R}^n \to \mathbb{R}^m$ with $$\left\lVert g(x) \right\rVert_2 \leq \ln(1+\left\lVert x \right\rVert_2^2) \text{ for all x}\in \mathbb{R}^n $$ How can I prove that $g$ is totally ...
0
votes
0answers
35 views

For which $\alpha \geq 1$ is $\left\lVert x \right\rVert^\alpha$ differentiable?

Let $\left\lVert \cdot \right\rVert_\infty$ be a norm on $\mathbb{R}^n$. How can one find out, for which $\alpha \geq 1$ the image $f$ with $f(x) := \left\lVert x \right\rVert^\alpha$ is totally ...
0
votes
1answer
10 views

norm of element in equivalent class in quotient space

If we have a quotient space $E\backslash L_0$ where $E$ is a linear normed space and $L_0$ it's subspace the norm of an element $L$ in $E\backslash L_0$ is defined as $$\lVert L\rVert = \inf_{x \in L}{...
1
vote
1answer
29 views

Calculating operator norm $\|Ax\|$

a) Consider $(\mathbb{R}^n, \|.\|_\infty)$ and $(L_b(\mathbb{R}^n, \mathbb{R}),\|.\|)$ which is the space of all linear and bounded functions from $\mathbb{R}^n \to \mathbb{R}$ associated with the ...
0
votes
0answers
14 views

Inner product between Gaussian radial basis functions

Let $\phi : \mathbb{R}^n \rightarrow \mathbb{R}$ be the Gaussian radial basis function: $$\phi(x) = \exp(-|x|^2)$$ Let $$f_i(x) = \phi{\left(\frac{x - \mu_i}{\sigma_i}\right)}$$ I'm computing the ...
1
vote
1answer
42 views

Inequality in $L^2$

Let $u,v\in L^2(\mathbb{R}^d)$, I want to prove the following ineqality $$\|u\|^2_{L^2(\mathbb{R}^d)}\ge a \|v\|^2_{L^2(\mathbb{R}^d)}-\|u-v\|^2_{L^2(\mathbb{R}^d)}$$ for all $u,v\in L^2(\mathbb{R}^...
0
votes
1answer
38 views

Frobenius and operator-2 norm

I have been studying about norms and for a given matrix A, I haven't been able to understand the difference between Frobenius norm $||A||_F$ and operator-2 norm $|||A|||_2$. Can someone help me ...