The norms tag has no usage guidance.

**4**

votes

**2**answers

49 views

### Distribution of the RKHS norm of the posterior of a Gaussian process

In a classical noisy regression setting, let $\big(f(x)\big)_{x\in\cal X}$ be a centered Gaussian process of covariance $k$ on a compact $\cal X$, and $\mathcal{F}_n$ be the filtration generated by ...

**0**

votes

**0**answers

149 views

### Writing integers in ring of integers of number fields

Given $a,b\in\Bbb N$, we can write $a=a_tb^t+a_{t-1}b^{t-1}+\dots+a_1b+a_0$ where $t=\lceil\log_ba\rceil$ and $a_i<b<a$.
(1) Supposing if $b\in\mathcal{O}_K$ where $\mathcal{O}_K$ is ring of ...

**1**

vote

**1**answer

75 views

### Cea's lemma and norms

I would like your help understanding this article.
Page 239 (3.2 A priori error estimates), I am quickly getting lost because of the type of norm that is always changing.
Things I do not ...

**1**

vote

**1**answer

64 views

### A bound on the $\mathrm{L}^p$ norm in terms of the $\mathrm{L}^2$ norm in periodic Sobolev spaces

(The question was originally posted on Math StackExchange.)
Preliminaries: Given ${s \geq 0}$, let ${\mathrm{H}_P^s}$ denote the ${\mathrm{L}^2}$-based fractional-order Sobolev space of ...

**0**

votes

**1**answer

76 views

### Comparison of Lp norm of matrix and its entry wise norm. [closed]

I need to know the relation between operator norm of a matrix i.e. $ \Vert A\Vert_p$ for case of p=1 and 2 and its entry wise Frobenius norm $ \Vert A\Vert_F$.

**2**

votes

**0**answers

47 views

### Algebraic independence in normed spaces

A set of $n$ points in $\mathbb{R}^2$ is algebraically independent over $\mathbb{Q}$ if there is no polynomial dependency among the $2n$ coordinates.
A result (Lemma 3.3) from "Globally linked pairs ...

**0**

votes

**0**answers

44 views

### Does such a function exists that approximates l1 norm by l2 norm for proportional data

A proportional vector $x\in \mathbb{R}_{+}^{d}$ is defined as (1) $x_{i}\geq 0$ ;(2) $\sum_{i=1}^{d}x_{i}=1$; $d$ is independent of anything else.
For two arbitrary proportional vector $x$ and $y$, ...

**5**

votes

**1**answer

172 views

### Under what conditions a linear automorphism is an isometry of some norm?

Assume $V$ is a finite-dimensional vector space over $\mathbb{R}$, and $T: V \to V$ is a (linear) isomorphism.
When is it possible to construct a norm on $V$
making $T$ an isometry?
...

**0**

votes

**0**answers

56 views

### The 2-norm of a positive circulant matrix

Define a circulant matrix $A$ for complex numbers $a_1, a_2, ..., a_n$ as follows:
$$
\text{circ}(a_1,\ldots,a_n)=
\left[ \begin{matrix}
a_1& a_2 & \cdots & a_{n-1} & a_n \\
a_n& ...

**11**

votes

**2**answers

440 views

### which norms can be realized as operator norms?

Assume $(V,∥∥_V),(W,∥∥_W)$ are both finite dimensional normed spaces. We have the induced operator norm on ${\rm Hom}(V,W)$.
It turns out that the operator norm is induced by an inner product iff ...

**15**

votes

**3**answers

418 views

### An inequality for two independent identically distributed random vectors in a normed space

Suppose that $X$ and $Y$ are independent identically distributed random vectors in a separable Banach space $B$. Does it always follow that $E\|X-Y\|\le E\|X+Y\|$?
Some background information on ...

**2**

votes

**1**answer

133 views

### When do Borel $\sigma$-algebras generated by the total variation norm and the weak* topology coincide?

I am almost certain that I read somewhere that the following is true, but I cannot seem to locate the reference. I would be most appreciative if someone could point me to a reference. The result was ...

**1**

vote

**0**answers

56 views

### Optimization with random matrix

Consider $J$ a random matrix of size $n\times n$ with i.i.d. Gaussian entries $J_{ij} \sim \mathcal{N}(0,\sigma^2/n)$. Let $f(x)=tanh(x)$, and for $x\in\mathbb{R}^n$, $f(x)$ denotes the vector where ...

**2**

votes

**1**answer

89 views

### Functions that are easy to compare to a norm

Let $X$ be a subset of $\mathbb{R}^d$, let $\|\cdot \|_p$ be a norm with $1\leq p\leq\infty$, and let $f:\mathbb{R}^d\to\mathbb{R}$ be a function. I'm trying to find examples of $X$, $p$, and $f$ for ...

**1**

vote

**0**answers

40 views

### Upper semicontinuity in C(X)-algebras. Quotient norm question

upper semicontinuity in C(X)-algebras
In the 5th paragraph of this post, I don't understand why there exists a vector b satisfying $||a+b||_A < ||q_x (a)||_A(x)$
By the definition of the quotient ...

**15**

votes

**1**answer

865 views

### How many values determine a norm?

It is well known that for a bilinear form over an n-dimensional vector space, $n^2$ values (on all pairs of basis-vectors) determine it uniquely.
How many values do we need to specify in order to ...

**2**

votes

**1**answer

134 views

### Equivalence of entrywise 1-norm and Schatten-1 norm

Let $A \in \mathbb{R}^{m\times n}$ and $\|A\| = \sum_{i, j} |A_{i,j}|$.
I am looking for constants $\alpha, \beta \in \mathbb{R}$ such that
$\alpha \|A\| \leq \|A\|_* \leq \beta \|A\|$
The function ...

**9**

votes

**2**answers

229 views

### A “quadratic” triangular inequality

In a Euclidian space (Hermitian as well), say $\ell^2_n$, the following inequality holds true
$$(QI)\qquad |b|\cdot|c-a|\le|c|\cdot|a-b|+|a|\cdot|b-c|,\qquad\forall a,b,c\in\ell^2_n.$$
In other words, ...

**3**

votes

**1**answer

108 views

### A norm description for singular matrices

For $n>2$, are there norms $\parallel.\parallel_{a}$ and $\parallel.\parallel_{b}$ on $M_{n}(\mathbb{R})$ with the following property:
$A\in M_{n}(\mathbb{R})$ is singular if and only if ...

**7**

votes

**1**answer

221 views

### Proving a certain $ C^{*} $-algebraic inequality

Let $ A $ be a non-unital $ C^{*} $-algebra. Is there an ‘elementary’ way to prove, for all $ (a,\lambda) \in A \times \mathbb{C} $, the inequality
$$
|\lambda| \leq \sup_{b \in A, ~ \| b \| \leq 1} ...

**20**

votes

**6**answers

886 views

### Under what conditions $\|x-y\|=n\iff\|f(x)-f(y)\|=n.$ for $n\in\mathbf{N}$ implies isometry?

Let $X, Y$ be normed space and $f:X\to Y$ be a mapping. Assume that for all $n\in\mathbf{N}$, $$\|x-y\|=n\iff\|f(x)-f(y)\|=n.$$
Under what conditions this map will be an isometry?
Thanks

**3**

votes

**0**answers

118 views

### Norm condition in a Banach lattice

Consider the following "condition (J)" on the norm of a (real or complex) Banach lattice $E$: whenever $x$ and $y$ are disjoint (i.e., $|x|\wedge |y|=0$) then
$\|x+y\|+\|x-y\|=2\|x\|+2\|y\|$.
...

**4**

votes

**1**answer

96 views

### $2$-D Hlawka inequality

The classical counter-example to Hlawka inequality
$$|a+b|+|b+c|+|c+a|\le|a+b+c|+|a|+|b|+|c|$$
is the $l^\infty$-norm in dimension $3$, with vectors
$$a=\begin{pmatrix} 1 \\ 1 \\0 \end{pmatrix},\quad ...

**0**

votes

**0**answers

38 views

### Estimate bounds on Minkowski distance from point to a segment in Lp space

Assumptions
Let
$L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric),
$a,b$ be arbitrary $n$-dimensional points,
$c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...

**0**

votes

**0**answers

67 views

### Best s-term approximation and unit balls in weak $\ell^p$ norm

In the book "A Mathematical Introduction to Compressive Sensing" by Foucart and Rauhut there is the following asymptotic estimate at page 332, equation (11.1):
$$\sup_{\mathbf{x}\in B^N_{r,\infty}} ...

**1**

vote

**0**answers

44 views

### Modified Orthonormal Procrustes Problem

In the general orthonormal Procrustes problem, we want to find an orthonormal matrix $C$ to minimize $\|Y-XC\|_F^2$, where $Y$ is a known $n\times q$ matrix, $X$ is a known $n \times m$ matrix, and ...

**9**

votes

**3**answers

471 views

### Pathological product space norm

Let $X$ and $Y$ be two normed vector spaces and $n(\cdot, \cdot)$ be any norm on $\mathbb{R}^2$. Is it always possible to define a norm on the product vector space $X \times Y$ as $||(x, y)||_{X ...

**5**

votes

**1**answer

150 views

### On the induced matrix norm $\| \cdot \|_{2,\infty}$

The induced norm of the matrix $A$ as a map from $(\mathbb R^n , \| \cdot \|_p)$ to $(\mathbb R^m, \| \cdot \|_q)$ is given by
$$ \| A \|_{p,q} = \sup_{x\in\mathbb{R}^n\setminus \{0\}} ...

**1**

vote

**0**answers

138 views

### RKHS norm and posterior of Gaussian process

In Srinivas et al (2010) [appendix B], the authors claim the following "easy to see" property relating the norm of a function in a RKHS induced by a kernel $k(\cdot,\cdot)$, and its norm in the RKHS ...

**7**

votes

**1**answer

430 views

### Counting norms on an infinite dimensional vector space

It is known that whenever E is a finite dimensional real vector space, there is only one norm on E up to equivalence (actually one non discrete vector space topology).
Is it known what happens when E ...

**1**

vote

**0**answers

222 views

### Bounds on the effect of a matrix product on the Frobenius norm

I was wondering if there was a way to put upper and lower bounds on the Frobenius norm of a matrix product in relation to the Frobenios norm of one of the individual matrices, i.e,
...

**2**

votes

**0**answers

49 views

### Image of the typenorm contains the squares

I am having a look at the paper Explicit CM-theory for level 2-structures on abelian surfaces by Bröker, Gruenewald and Lauter, and there is an argument which I don't understand. The main reason being ...

**0**

votes

**0**answers

78 views

### Characterize the set of roots of cubics with certain properties

Let $P(x)\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $3$. Suppose that $\alpha_1, \alpha_2, \alpha_3$ are roots of $P(x)$. For what such $P(x)$ is it the case that the ring of integers ...

**0**

votes

**1**answer

116 views

### Relation between the subordinate norm and the spectral radius of a matrix

Let's define the following subordinate norm of a $(NM \times NM)$ matrix A norm as follows
\begin{eqnarray*}
||A||_{2,b} = \mathrm{max}_{x \in \mathbb{C}^{NM}} \left \{ \frac{||A x||_b}{||x||_2} ...

**2**

votes

**0**answers

61 views

### Reducing $\ell_1$ norm of non-full-rank matrices

I have two matrices ${\bf{X}}_{p\times r}$ and ${\bf{Y}}_{r\times q}$ with $r<\min(p,q)$. Matrix ${\bf Y}$ does not have full row rank (i.e., rank$({\bf Y})<r$). Can I build two new matrices ...

**0**

votes

**2**answers

173 views

### Uniqueness of solution of a nonconvex optimization problem

What conditions need to be hold for a nonconvex optimization problem to have a unique solution?
Specifically, I have the following minimization problem that I'd like to know whether it has a unique ...

**2**

votes

**0**answers

57 views

### König-Wittstock norm on Banach space

Let $(E,\Vert.\Vert)$ be a real Banach space and $\ell\ne 0$
an non-continuous linear form on $E$.
Let $a\in E$ be such that $\ell(a)=1$.
König-Wittstock [Non-equivalent complete norms
and would-be ...

**2**

votes

**0**answers

50 views

### Invexity of the $L_2$ norm

I have the following function:
$ f({\bf A,b}) = \| {\bf y - XAb} \|_2^2$
where ${\bf y}_{n \times 1}$ and ${\bf X}_{n \times p}$ are fixed, and ${\bf A}_{p \times r}$ and ${\bf b}_{r,1}$ are the ...

**0**

votes

**2**answers

308 views

### Convexity of the Frobenius norm of the product of two matrices

I have the following function for two matrices ${\bf A}$ and ${\bf B}$:
$f({\bf A}, {\bf B}) = \| {\bf Y - XAB} \|_F^2 = trace\{({\bf Y - XAB)}^T({\bf Y - XAB)}\}$
where matrices ${\bf X}_{n \times ...

**2**

votes

**1**answer

94 views

### Approximation with a rank-$1$ matrix

Given a matrix $A$ (generally speaking, complex and non-square), I want to find an identically-sized matrix $D$ with ${\rm rk} D\le 1$ to minimize the induced operator norm $\|A-D\|_2$. From the ...

**1**

vote

**2**answers

159 views

### relation between of uniformly rotund in every direction and uniformly rotund and locally uniformly rotund

The norm of a Banach space $X$ is said to be uniformly rotund in every direction if $$\lim_{n→∞}\|x_n−y_n\|=0$$ whenever $$x_n,y_n∈SX$$ are such that $$\lim_{n→∞}\|x_n+y_n\|=2$$ and there is a $z∈X$ ...

**1**

vote

**3**answers

215 views

### What is the most ``diverse'' $k$-subset of $[0, 1]^m$?

Given a non-negative integer $m$, let $\Omega_m$ denote the set of vectors $\omega = (\omega_1, \dots, \omega_m) \in [0, 1]^m$ such that $\sum_i{\omega_i} = 1$.
The set $\Omega_m$ is together with a ...

**10**

votes

**3**answers

513 views

### Are norms intrinsically $\mathbb{R}$-valued?

Another way of phrasing this: are there any viable definitions of something which is norm-like but whose range is in a linearly ordered rig (for example) rather than $\mathbb{R}$?
I have searched a ...

**6**

votes

**2**answers

334 views

### $\|T\|_2 \le \sqrt{\|T\|_1\|T\|_\infty}$

Let $T$ be a linear operator acting on a finite-dimensional real or complex
vector space. As a direct consequence (or rather a particular case) of the
Riesz-Thorin theorem, we have
$$ \|T\|_2 \le ...

**2**

votes

**1**answer

94 views

### Bound on the 2-norm of a “special” matrix

Let $S\in\mathbb{R}^{n\times n}$ be such that $\|S\|_2\leq 1$, $P\in\mathbb{R}^{n\times m}$, $m<n$, with orthogonal columns ($P^TP=I$) so that $PP^T$ and $I-PP^T$ are orthogonal projectors, and ...

**10**

votes

**1**answer

711 views

### On bi-invariant metrics on groups

I'm looking for a quick, snap-your-fingers proof of the following result:
A continuous length metric on $\mathbb{R}^n$ that is invariant under translations comes from a norm.
To be clear about ...

**3**

votes

**1**answer

269 views

### Estimate infinity norm with Lp and W1p norm

Let $p \in [1,\infty)$. Does there exist $C>0$ such that for every $f \in W^{1,p}([0,1],\mathbb{R})$ we have
$$\|f\|_{L^\infty}\leq C\|f\|_{L^p}^{1-\frac{1}{p}}\|f\|_{W^{1,p}}^{\frac{1}{p}}?$$
My ...

**5**

votes

**0**answers

277 views

### Symmetric matrices with $\rho(A)\gg\|A\|_\infty$

For a symmetric real matrix $A$, denote by $\rho(A)$ the spectral radius of $A$, and by $\sigma(A)$ the largest absolute row sum of $A$; that is, $\sigma(A)=\max_i \sum_j |a_{ij}|$, where $a_{ij}$ are ...

**2**

votes

**0**answers

172 views

### Norm bound of a complex resolvent

A well known result by Varah states that if $A$ is a strictly diagonally dominant matrix of dimension $n$, then
$\|A^{-1}\|_{\infty} \le \max_i\frac{1}{|a_{kk}|-\sum_{j \neq k}|a_{kj}|}$, where the ...

**5**

votes

**1**answer

250 views

### An inequality involving the spectral norm of a complex matrix

Let $A,B \in {M_n}(R)$ be real $n \times n$ matrices and let matrices $|A|$ and $|B|$ contain the absolute values of the elements of $A$ and $B$ respectively. Construct the complex matrices $C = A + j ...