Questions tagged [norm]

This tag is for the questions relating to the norm which is a function on a vector space $X$ that generalizes notion of length of vector in general vector spaces. A vector space $~X~$ with a distinguished norm is called a normed space.

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2
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2answers
43 views

What is the benefit of defining a positive norm for vectors?

I read that the reason we have the property $\langle A|B\rangle=\langle B|A\rangle^*$ is to make define a positive norm with the formula $\langle A|A\rangle$. Though I do not understand how having ...
7
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0answers
66 views

For a normed vector space, is $\|x-y\| \leq \|x\|+\|y\|$ true?

I have a question about an inequality in normed vector spaces and I want to know if my proof is correct. Claim: Let $X$ be a normed vector space. Then \begin{equation} \|x-y\| \leq \|x\|+\|y\|\end{...
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0answers
36 views

What is the logic of the following normalization formulas?

Let's say we have two vectors $a$ and $b$. My question is what would be the purpose meaning of the following formulas : $$\dfrac{\Vert a - b \Vert}{\Vert a + b \Vert}$$ and, $$\dfrac{a b}{\Vert a\Vert ...
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0answers
67 views

Bounding operator $I^{\alpha} f(t) - \frac{1}{\Gamma (\alpha)} \int_0^t(t-\tau)^{\alpha - 1}f(\tau)d\tau.$

Let $p \in [1, \infty], \alpha > 0, T > 0$ and define $$I^{\alpha} f(t) - \frac{1}{\Gamma (\alpha)} \int_0^t(t-\tau)^{\alpha - 1}f(\tau)d\tau,$$ where $\Gamma$ is Gamma function. Show that $I^{\...
1
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1answer
36 views

Is the spectral norm of $I_n-1v^T$ bounded by $\sqrt{n}$?

Let $I_n$ be $n$-dimensional identity matrix and $v$ be a stochastic vector, i.e., $v$ is non-negative and $v^T{\bf 1}=1$, where ${\bf 1}=[1,1,...,1]^T$. I wonder if the spectral norm of the matrix $...
0
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1answer
40 views

Relation between norms and inner product

I'm trying to show that for $\mathbf{x} \in \mathbb{C}^N$ and $\mathbf{A} \in \mathbb{C}^{m\times N}$ that $$ \|\mathbf{Ax}\|_2^2-\|\mathbf{x}\|_2^2=\langle(\mathbf{A}^\ast \mathbf{A}-\mathbf{I})\...
0
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1answer
20 views

Scalar derivative of vector norm

Can someone check my math here? I feel like this should be a very simple problem, but I can't seem to confirm this by searching. What is the derivative of a vector norm? I'm referring to the usual ...
0
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1answer
46 views

Proving the triangle inequality for a function on matrices

Let $V$ be a $m$-dimensional vector space over the set of $n \times n$ matrices. For instance a vector would be $$ v=(M_1,M_2,\dots, M_m) $$ where each $M_i$ is a $n\times n$ matrix. I now define ...
0
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1answer
21 views

Does $T_n$ converge to the identity operator $I$ by the operator norm in $C[0,1]?$

Consider a sequence of operators $T_n$ :$C[0,1]\rightarrow C[0,1]$ given by the formula $$(T_nx)(t)=x(t^{1+\frac{1}{n}}),\ t\in [0,1], \ n\in \mathbb{N}.$$ Does $T_n$ converge to the identity operator ...
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0answers
26 views

Inner product of vector of matrices - I have the norm

Let G be the set of all 4 x 4 matrices with positive determinant. I define v as a vector such matrices. The norm of v is defined as: $$ ||v||^2=\sqrt{\sum_{g \in v}\det g} $$ What is the inner product?...
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46 views

Are all matrix norms equivalent?

I have shown that in $\mathbb{R}^n$all vector norms are equivalent. Does the same hold also for all matrix norms $\|\cdot \|:\mathbb{R} ^{n\times n} \rightarrow \mathbb{R} ^2$?
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2answers
20 views

Prove that the Frobenius norm is invariant under orthonormal projection

Assume I can express a rank-deficient, $N\times N$ symmetric covariance matrix $\Sigma$ as \begin{equation} \Sigma=\mathbf{USU}^\top \end{equation} where $\mathbf{U}$ is an $L\times N$ orthonormal ...
0
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1answer
36 views

Norm of a vector component (considering an orthogonal basis) is always lesser than or equal to the norm of the entire vector in $\mathbb R^n$ [closed]

Suppose $\mathbb R^n$ (with the usual dot product as inner product) is equipped with some arbitrary norm $||\cdot||$. Now let's say $\mathbf x = (x_1, x_2, \ldots, x_n)$ is a vector in $\mathbb R^n$ (...
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1answer
19 views

Matrix Derivative of F-norm with Hadamard Product

I'm trying to solve $\nabla_X \| A \odot(B-X^\top C) \|_F^2$, but I don't know how to solve this... Could anyone help? Thank you in advance for any help you can provide.
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0answers
22 views

Bounded linear functional on a Hilbert space.

Let $H$ be a complex Hilbert space with orthonormal basis $ \{e_n,n=1,2,\cdots\}$ and $f$ be a linear functional on $H$ defined by $$f(x) =\sum_1^\infty \langle x, e_n\rangle \frac{1}{n}.$$ How to ...
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0answers
60 views
+50

Minimize $x^*(A+A^*)x$ such that $x^*A^*Ax=1$ and $x^*x=1$

Given $A\in\mathbb{C}^{n\times n}$, such that it has singular values larger than $1$ and smaller than $1$, \begin{array}{ll} \underset{x\in\mathbb{C^n}}{\text{minimize}} & x^*(A+A^*)x.\\ \text{...
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0answers
12 views

Sommerfeld radiation condition [duplicate]

could I ask you please about Sommerfeld radiation condition (for example link: https://en.wikipedia.org/wiki/Sommerfeld_radiation_condition)? What does mean derivative with respect to Euclidean norm? ...
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0answers
42 views

Solving convex optimization by ADMM using python

I would like to solve the following optimization problem with ADMM. Then I implemented ADMM with python. But it doesn't work. The following equation is the problem I want to solve. $$mim\frac{1}{2}\...
0
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1answer
37 views

find norm $T:C[0,\pi]\to C[0,\pi]$ by $(Tx)(t)=\int ^t_0 \cos(t-s)x(s) ds$

Find the norm of following opeartor $$T:C[0,\pi]\to C[0,\pi] \text{ by } (Tx)(t)=\int ^t_0 \cos(t-s)x(s) ds$$ \begin{align} \| Tx\|_{\infty }&=\sup _{t\in[0,\pi]} \int ^t_0 \cos(t-s)x(s) ds\\ &...
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0answers
36 views

Hilbert space with a “norm” that does not respect the triangle inequality? (attempt 2)

This is my second attempt, following the first question here: Hilbert space with a "norm" that does not respect the triangle inequality? I believe this inner product produces a Hilbert space,...
0
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1answer
47 views

Is there a norm that can specify an unique vector $x$?

Assume that we have a vector $x$ with dimension $3$. The value of $x = {4, 2, 8}$. My question is if there is a norm that can generate one value of $x$ that are unique compared to other samples of $x$....
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0answers
30 views

How to prove $\| (1 + 2 \tau)[I - \tau (D - \mathrm{diag}(1 + \mathbf u.^2))]^{-1}\mathbf u^n\|_{l^\infty} \leq 1$?

Consider a PDE of the form \begin{equation} u_t = \Delta u + u - u^3, \quad x \in[a, b] \end{equation} with periodical boundary condition, which is equivalent to \begin{equation} u_t = (\Delta - 1- ...
1
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1answer
23 views

Can I use $||J||_2$ as a “gradient descent” for the system $J = b - Ax$?

One quick question. Let's assume that I want to solve $Ax = b$ but I want to do that in a special way. My idea is that I first find the difference between $b$ and $Ax$, we call it $J$. $$J = b-Ax$$ ...
0
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1answer
28 views

Why minimize squared L2 norm and not only the L2 norm?

I'm studying Inverse Problems and usually, they minimize the squared of the L2 norm($L_2, L_0, L_ \infty$), why don't minimize only the norm? if the goal is to have a measure of the distance between 2 ...
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0answers
14 views

Extension of function in Hölder class outside of compact set

Let $K_1 \subsetneq K_2\subseteq \mathbb{R}^n$ be two connected and compact subsets. Suppose we have $\beta > 0$ and a function $f: K_1 \rightarrow \mathbb{R}$ satisfying (using the Multi-Index-...
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1answer
18 views

derivation of L2 norm of matrix formula [closed]

consider a matrix $A \in \mathbb{R}^{m\times n}$, its L-2 norm is defined as the maximum eigenvalue of $A^T A$. I tried to derive this from $||A||_2=\max_{x\neq 0}\frac{||Ax||_2}{||x||_2}=\max_{x\neq ...
3
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2answers
68 views

What does $ \|f\|_{K^{\infty}} $ mean in this context?

The following is from the paper The Modern Mathematics of Deep Learning page 23. For the sake of simplicity, I've hidden the details. The last line is $$\min _{f \in \mathcal{H}_{K} \infty}\|f\|_{K^{\...
1
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1answer
32 views

On compatible and natural norms (Exercise 2.7.12b in Kreyszig's Functional Analysis text)

$ \newcommand{\nc}{\newcommand} \nc{\C}{\mathbb{C}} \nc{\F}{\mathbb{F}} \nc{\R}{\mathbb{R}} \nc{\a}{\alpha} \nc{\n}[1]{\left \Vert #1 \right \Vert} \nc{\abs}[1]{\left \vert #1 \right \vert} \nc{\set}[...
0
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1answer
36 views

Does $f(t,x)=tx^2-tx$ satisfy Lipschitz condition?

I have function $f: [0,T] \times \mathbb{R} \rightarrow \mathbb{R}$, $T \in \mathbb{R_+}$ $$f(t,x)=tx^2-tx$$ I want to check if it satisfies Lipschitz condition and I got $$|f(t_1, x_1)-f(t_2, x_2)|=|...
0
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0answers
32 views

Why is this true: $\lVert a \rVert _1 \geq s^{1/2} \lVert b \rVert_2$ [closed]

Under the condition that the two vectors a and b : $\max|a| \geq \min|b|$ are s-sparse How do you prove that: $\lVert a\rVert_1 \geq s^{1/2} \lVert b \rVert_2$
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0answers
53 views

Expected norm of matrix product with random unit vector

Let $S_d = \{x \in \mathbb{R}^d : \|x\| = 1\}$ be the unit sphere in $\mathbb{R}^d$. Let $A \in \mathbb{R}^{n \times d}$. Note that \begin{align} \max_{x \in S_d} \|A x\| &= \max \sigma(A) &...
2
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2answers
67 views

Find a $\bar v$ with $\|\bar v\|_\infty=\alpha$ such that $\langle u \, , \bar v \rangle = - \|u\|_1\|v\|_\infty = -\alpha\|u\|_1$

The original question is: Find a $\bar v \in \mathbb{R}^n$ with $\|\bar v\|_\infty=\alpha$ and $\alpha>0$ express it in terms of $u \in \mathbb{R}^n\setminus\{0\}$ and $\alpha$ such that, $\...
0
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0answers
33 views

Sobolev seminorm related to affine mapping

the context of my question is finite element theory here, but my question is purely mathematical. Let $F$ be an affine mapping from a reference triangle $\hat{T}$ to a general triangle $T$, call it $F:...
0
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0answers
21 views

An inequality about $H^{1}$ Space

Consider $f \in C^{1}(\mathbf{R})$.$ U_{1},U_{2}: [0,1] \rightarrow \mathbf{R}$ are $H^{1}(0,1)$ functions. My question is how to prove that:$\vert\vert f(U_{1})-f(U_{2}) \vert \vert_{H^{1}(0,1)} \leq ...
1
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1answer
57 views

Is my proof of this operator norm correct?

Let $\alpha \in \ell^\infty$ and $T_\alpha:\ell^p \rightarrow \ell^p$ $(1\leq p \leq \infty)$ given by \begin{equation} T_\alpha(x)=(\alpha_1x_1,\dots,\alpha_nx_n,\dots). \end{equation} Note that $||...
0
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1answer
36 views

bounds between $l_1$ and $l_2$

It is known that in finite-dimensional spaces $l_1, l_2$ and $l_\infty$ norms are equivalent. That is, there exists constants $C_1,C_2 > 0$ such that $$\forall x \in \mathbb{R}^n\ C_1||x||_1 \le ||...
2
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0answers
23 views

Let $m\in C[a, b]$. Consider on $(C[a, b], ||m||_∞)$ the Multiplication operator $A:C[a, b]\to C[a, b], Af=mf$. prove that $||A||=||m||_∞$.

Let $m\in C[a,b]\;$. Consider on $\big(C[a,b],\Vert m\Vert_{\infty}\big)$ the multiplication operator $\;A:C[a,b]\to C[a,b]\;,\;Af=mf\;$. Prove that $\;\Vert A\Vert=\Vert m\Vert_{\infty}\;$. My try: ...
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0answers
15 views

Direct sum of two bases of Banach spaces

Consider $X = l_p \oplus l_q, 1 \leq p < q < \infty,$ and let $\mathcal{B} = (e_n)_{n=1}^\infty$ be the direct sum of the natural unit vector bases of the two spaces. That is, in our basis we ...
0
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1answer
47 views

What does the energy norm $\sqrt{err^T*A*err}$ tell?

What does the energy norm $\sqrt{err^T*A*err}$ tell? Where $A$ is the matrix that would be used in solving the problems linear system of equations $Au=b$. $err=u_{approx}-u_{exact}$ Particularly, I ...
0
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0answers
10 views

Full rank lattices and norms

There is a sentence in Micciancio's Lattice lectures that says when we bound the minimum distance of a lattice using the convex body theorem for norms other than the Euclidean norm we need to assume ...
1
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1answer
48 views

Is the Euclidean norm the only norm that admits “non-reflective” isometries?

Let $\|\cdot\|$ be an arbitrary norm on $\mathbb R^n$, and suppose there exists an isometry $T:\mathbb R^n\to\mathbb R^n$ that is not simply a composition of reflections along some of the $n$ axes. ...
4
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0answers
59 views

Is the Euclidean norm canonical?

In the spirit of this and this question, I'm interested in the motivation for defining the Euclidean norm in $\mathbb R^n$ to be $\|x\|=\sqrt{\sum_ix_i^2}$. Of course, Euclidean geometry provides a ...
1
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2answers
50 views

All possible norms on a finite-dimensional vector space?

Let $X=\mathbb K^n$, where $\mathbb K=\mathbb R$ or $\mathbb C$. I have seen proofs that the functions $$\|x\|_p:=\sqrt[p]{\sum_i|x_i|^p},\qquad p\in[1,\infty]$$ are all norms. (The $p=\infty$ case ...
1
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0answers
29 views

Vectors such that $\lVert x + y \rVert_p^p = \lVert x \rVert_p^p + \lVert y \rVert_p^p$

We fix $1 \leq p < 2$. What are the couple $({x},{y})$ of vectors in $\mathbb{R}^2$ (or more generally in $\mathbb{R}^n$) for which the following equality holds \begin{equation} \label{eq:here} \...
0
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0answers
32 views

Functional bound on $\Bbb R \times \{0\}$

Let's define functional on $\Bbb R \times \{0\}$ as: $$f((x, 0)) = -2x$$ I want to judge whether this functional is bounded assuming that on $R \times \{0\}$ we consider norm $\|x\|_1 = \sum_{i=1}^\...
0
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1answer
18 views

Real, symmetric matrices $A, B \ge 0$ with $A$ positive definite, must we have $\lVert (A +B)^{-1}B \rVert_{\text{op}} \le 1$?

If $A$ is positive definite and $B$ is positive semi definite, both symmetric real matrices, with $A - B$ positive semi-definite, must we have $\lVert (A+B)^{-1}B \rVert \le 1$ in the operator norm? I ...
1
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2answers
68 views

Norm inequality for positive semidefinite matrices

Suppose $A, B\ge0$ are positive semidefinite matrices on the complex field, is it true that $$\Vert A^2 +B^2 \Vert \le \Vert A + B \Vert^2,$$ for the spectral norm? I have tried numeric tests and the ...
0
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0answers
8 views

Lipschitzness of the fractional powered euclidean norm .

Can someone help me to provide an upper or lower bound to the following? $$ \|\nabla_xf(x_1)\|^{\alpha}-\|\nabla_xf(x_2)\|^\alpha\leq?$$ where $0<\alpha<1$. I prefer a bound in terms of $\|\...
-2
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1answer
42 views

Show that the operator norm $\sum$ defined as $\sum(\lbrace{x_n\rbrace}_{n=0}^\infty) = \sum_{n=0}^\infty \lbrace{x_n\rbrace}$ is unbounded.

Suppose $\|\lbrace{x_n\rbrace}_{n=0}^\infty \|_{\text{sup}} = \text{sup}\lbrace{|x_n|\rbrace}_{n=0}^\infty $. Let $\sum$ be a linear operator from the $A \to \mathbb{R}$ defined as $\sum(\lbrace{x_n\...
4
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0answers
78 views

“Canonical” norm on a real finite-dimensional unital associative division algebra?

Let $\mathcal{C}$ denote the category of unital associative finite-dimensional division $\mathbb{R}$-algebras. (As a full subcategory of that of unital $\mathbb{R}$-algebras.) For $A\in \mathcal{C}$ ...

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