The norms tag has no usage guidance.

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84 views

### Norm vs A-norm in non-Archimedean Functional Analysis

Let $K =(K,| \cdot |)$ be a non-Archimedean valued field.
Let $E$ be a $K$-vector space. A norm on $E$ is a map $||\cdot||:E\to[0,\infty)$ such that:
$||x||=0$ if and only if $x=0$,
$||\lambda ...

**8**

votes

**3**answers

495 views

### Can the topological algebra of analytic functions be endowed with a norm that defines the natural topology?

Right, so in my research in complex analysis I was puzzled by this question which may have a simple approachable answer that eludes me, but I am truly itching to find out and in need of it so I am ...

**1**

vote

**3**answers

113 views

### Norm of an operator formed using a unitary operator

Suppose, $ A $ is a unitary matrix in $ M_n(\mathbb{C}) $ given by $ (a_{i,j})_{1\le i,j\le n} $ which has the property that, for all the basis elements $ e_i $, $ Ae_i\ne |\lambda| e_j $ for all $i,j ...

**1**

vote

**1**answer

64 views

### Different Besov-Norm Definitions

first some notation: $\langle x\rangle=\sqrt{1+x^2}$, $P_{j}$ is the Littlewood Paley Projector and $P_{\leq0}$ corresponds to the small frequencies.
I have a the following definition of the Besov ...

**0**

votes

**1**answer

158 views

### How to calculate $N_{L/k}$(roots of unity)?

Suppose that $L/k$ is a Galois extension of number fields and that $G$
is the corresponding Galois group. Further, for $\frak p$ a prime ideal
of $\cal O$$_L$, let $K=L^{G(\frak p)}$, where [$L$ : ...

**23**

votes

**0**answers

478 views

### Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?

Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $||x|| = 1$ and $||Ax|| = ||A||$. The ...

**3**

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**0**answers

136 views

### Lenstra's integer programming algorithm: Finding a lattice point “near the center”

I have already posted this question on the mathematics forum, but I suspect the question needs more detailed knowledge than most users have; please excuse the duplicate post. Any help is greatly ...

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**0**answers

92 views

### Unit sphere of a norm is a submanifold implies the norm is smooth?

Let us call a norm on $\mathbb{R}^n$ smooth if its restriction $\| \cdot \|:\mathbb{R}^n\setminus \{ 0 \} \to \mathbb{R}$ is a smooth map.
Suppose the unit sphere of a norm $\| \cdot \|$ is an ...

**1**

vote

**1**answer

144 views

### Under what conditions can we put a complete norm on a linear subspace of a separable Banach space?

Question 1 Let $X$ a separable Banach Space and $Y\subset X$ linear subspace. When can we put a norm on $Y$ in such a way so that $Y$ is a Banach space?
Clearly if $Y$ is closed in the norm topology ...

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votes

**1**answer

40 views

### Ratio of sums of $\infty$-norms of vectors

For any vector $w\in\mathbb{R}^n,$ let $\|w\|:=\|w\|_\infty = \max_i |w_i|.$ Let $w\geq 0$ mean that $w$ is non-negative in each co-ordinate.
For $0<\alpha<\frac 14,$ I am interested in the ...

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vote

**2**answers

260 views

### Looking for (information about) long diamonds

I was given an open problem as a birthday present recently. While I can probably handle spoilers at this point, what I really want are literature and other references. Also acceptable would be ...

**8**

votes

**2**answers

297 views

### Constructing a function over a metric space through given points

Suppose there is a compact metric space $(X,\rho)$ and a Euclidean space $\mathbb{R}^n$.
There is a sequence of unequal points $\{x_1,...x_N\}$ in $X$ such that all metrics $\rho(x_i,x_j)$ are known ...

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**0**answers

118 views

### GPS calculations under $L^p$ norms

GPS calculations require finding a sphere externally tangent to
four given spheres, an
Apollonian problem
in $\mathbb{R}^3$.
The center of that fifth sphere is one of the $16$ possible solutions to
...

**2**

votes

**0**answers

112 views

### Minimize L-infinity norm with restrictions

I need to minimize the following L-infinity norm with respective to $\tau$. L-infinity norm of a matrix $A$ is defined as $\|A\| = max_{i,j}|a_{i,j}|$.
$$
min_{\tau} \| I -S(S+\tau)^{-1}\|
$$
$$
...

**0**

votes

**1**answer

202 views

### L-infinity-norm regularized proximity problem

I have a question:
$$\min_x {1\over 2} \|x-t\|^2 + \lambda \|x\|_\infty$$
where $t, \lambda$ are given constant.
I think this may be a classic problem? However, I didn't find closed form of its ...

**4**

votes

**2**answers

61 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 ...

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**0**answers

155 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

94 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 ...

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vote

**1**answer

71 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 ...

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votes

**1**answer

171 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$.

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votes

**0**answers

51 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 ...

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**0**answers

96 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 $y$, ...

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votes

**1**answer

208 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?
...

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**0**answers

81 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& ...

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votes

**2**answers

475 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 ...

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votes

**3**answers

459 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 ...

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votes

**1**answer

174 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 ...

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vote

**0**answers

71 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 ...

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votes

**1**answer

95 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 ...

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**0**answers

44 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 ...

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votes

**1**answer

952 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 ...

**3**

votes

**1**answer

207 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

285 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

119 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 ...

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votes

**1**answer

241 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} ...

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votes

**6**answers

939 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

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**0**answers

130 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
$\|x+y\|+\|x-y\|=2\|x\|+2\|y\|$.
...

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votes

**1**answer

104 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 ...

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43 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 ...

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**0**answers

48 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 ...

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**3**answers

523 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 ...

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votes

**1**answer

205 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\}} ...

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vote

**0**answers

195 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 ...

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**1**answer

557 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 ...

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**0**answers

353 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

52 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 ...

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**0**answers

79 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**

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**1**answer

145 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} ...

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62 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

202 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 ...