Questions tagged [norms]

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Bounding supremum norm in terms of gradient L2-norm using a Poincare-like inequality

Suppose $f$ is a Lipschitz continuous real-valued function over a bounded domain $\Omega \subset \mathbb{R}^d$ with smooth boundary, and let $\overline{f} := \frac{1}{|\Omega|}\int_\Omega f(x) dx$. Is ...
Greg O.'s user avatar
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Verifying the Cauchy behavior of a sequence

Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a non-empty subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^...
PPB's user avatar
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1 answer
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On a property for normed spaces

I asked this question on Math Stackexchange, but I didn't get an answer: https://math.stackexchange.com/questions/4881155/on-a-property-for-normed-spaces?noredirect=1#comment10410489_4881155 I came ...
Markus's user avatar
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Strong sub-differentiability of an equivalent strictly convex norm

First, we define the notion of strong sub-differentiability(SSD) of a norm on a Banach space $X$. The norm $\Vert \cdot \Vert$ of $X$ is said to be SSD if the one-sided limit $$\lim_{t \to 0+} \frac{\...
PPB's user avatar
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4 votes
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Computing preimage of element under norm map of quadratic extension of $2$-adic fields

Let $F$ be a $2$-adic field, i.e. a finite extension of the $2$-adic numbers $\mathbb{Q}_2$. Suppose that I have a quadratic extension $E = F(\sqrt{d})$ of $F$. Given a unit $\alpha \in \mathcal{O}_F^\...
Sebastian Monnet's user avatar
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168 views

Degree 6 Galois extension over $\mathbb{Q} $

Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...
Sky's user avatar
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Convergence of slice in an equivalent renorming

Let us consider $\ell_2$ space with $\Vert \cdot \Vert_2$ norm. Let us define a new norm equivalent to $\Vert \cdot \Vert_2$ norm as follows: $$ \Vert x \Vert_0 = \max \{ \Vert x \Vert_2, \sqrt{2} \...
PPB's user avatar
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2 votes
1 answer
145 views

Are these two norms on localized versions of $L^p_q$ equivalent?

$\newcommand{\RR}{\mathbb R}\newcommand{\diff}{\, \mathrm d}$ We fix $T \in (0, \infty)$ and $p, q \in [1, \infty)$. Let $\mathbb T$ be the interval $[0, T]$. Let $E$ be the space of all real-valued ...
Akira's user avatar
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Convexity property of an equivalent norm on $\ell_2$

Let us consider the space $\ell_2$ with an equivalent norm defined by $$ \Vert x \Vert = \max \{ \Vert x^{'} \Vert_2, \Vert x^{''} \Vert_2 \}, $$ where $x^{'}=(0, x_2, x_3, \cdots)$, $x^{''} = (x_1, 0,...
PPB's user avatar
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1 answer
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Norm of a matrix function of a vector $x$ [closed]

I have a matrix $$A(x) = \frac{-1}{(1+\|x\|_2^2)^{\frac{3}{2}}}xx^T + \frac{1}{(1+\|x\|_2^2)^{\frac{1}{2}}}I$$ I see that $$\|A(x)\|_2 \le 1 \ \forall x$$ But is $\|A(x)\| \le 1$ in general $\forall x$...
Shoeb's user avatar
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2 votes
1 answer
160 views

Does there exists an example of a Banach space that is compactly LUR; but not LUR

We know that a Banach space $X$ is locally uniformly rotund(LUR) if for $x, x_n \in S_X$ with $\Vert x+x_n \Vert \to 2$, we have $x_n \to x$. In the same case, if $(x_n)$ has a convergent subsequence, ...
PPB's user avatar
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1 answer
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Handling the $\ell^2$ norm of a matrix expression in a linear regression

I am reading a scientific article in which matrices are handled (which I do not use often). We consider a matrix $X\in\mathbb R^{n\times p}$ and a vector $y\in\mathbb R^n$. The authors show that the ...
Zach's user avatar
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Distribution of joint Gaussian conditional on their sum of squares

Given a random gaussian matrix $\mathbf{X}$ with zero mean matrix and covariance matrix $\mathbf{\Sigma}$, and two deterministic matrices $\mathbf{A}$ and $\mathbf{B}$. If I know the value of $\|\...
Sheperd Lv's user avatar
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Weighted version of the Gagliardo Nirenberg inequality

I'm searching for a weighted Gagliardo-Nirenberg inequality similar as in https://arxiv.org/pdf/1307.1363.pdf where the weight is a power of the last component. Is there an inequality of the form $$ \...
user99432's user avatar
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0 answers
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Lower & upper bound on the maximal component given the system of power sums

Given a non-negative vector $x=(x_1,x_2,\dots,x_n)\in\mathbb{R_{>0}^n}$ and $m\in\mathbb{N}$, construct a system of power sum symmetric polynomials (or norms, if you like) $$ \begin{cases} x_1+x_2+\...
Polylemma's user avatar
15 votes
2 answers
892 views

Higher-rank Archimedean valuations of $\mathbb{Q}$, does it exist?

I was reading the proof of Ostrowski's theorem, with an eye toward the Zariski-Riemann space (as well as adic space, Berkovich space, etc.) In the proof, the value group is always assumed to be in $\...
zxx's user avatar
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1 vote
1 answer
125 views

Smoothness of an equivalent norm

For an arbitrary set $\Gamma$, Day's norm on $c_0(\Gamma)$ is defined by $$ \Vert x \Vert = \sup \bigg \{ \bigg ( \sum_{k=1}^n 4^{-k} x^2(\gamma_k) \bigg )^{\frac{1}{2}} : (\gamma_1, \cdots, \gamma_n) ...
PPB's user avatar
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0 answers
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Convexity of an equivalent norm

Let $X=l_2$ with usual norm $\|\cdot\|_2$. We define a subspace of $X$ as $D=conv (B_{l_2} \cup B),$ where $B = \{ (x_n) \in l_2 : \sum_{n=1}^\infty \frac{n}{2} x_n^2 \leq 1\}$, conv is the convex ...
PPB's user avatar
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Is there $r>0$ such that the norm $[f]_r:=\sup_{n \ge 1} \|1_{B(y_n, r)} f\|_{L^p}$ is equivalent to $\|f \|:=\sup_{y \in Y}\|1_{B(y,1)}f\|_{L^p}$?

Below we use Bochner measurability and Bochner integral. Let $(Y, d)$ be a separable metric space, $\mathcal B$ Borel $\sigma$-algebra of $Y$, $\nu$ a $\sigma$-finite Borel measure on $Y$, $(E, |\...
Analyst's user avatar
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9 votes
2 answers
815 views

What norms can be "universally" defined on any real vector space with a fixed basis?

Let $V$ be a real vector space and let $B = (b_\lambda)_{\lambda \in \Lambda}$ be a basis. So every $v \in V$ can be written uniquely as a linear combination $$ v = c_{\lambda_1} b_{\lambda_1} + c_{\...
AlpinistKitten's user avatar
0 votes
0 answers
72 views

Property (H) in the dual norm

Consider the Hilbert space $l_2$ with an equivalent norm $$\Vert x \Vert = \max \{2 \Vert x \Vert_1, \Vert x \Vert_2 \},$$ where $\Vert x \Vert_1 =( \sum_{n=2}^\infty x_n^2 )^{\frac{1}{2}}$ and $\Vert ...
PPB's user avatar
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4 votes
0 answers
97 views

Frobenius norm bounds on exponentials of anti-Hermitian matrices

Suppose $X$ and $Y$ are two anti-Hermitian matrices satisifying $\|X\|, \|Y\| \leq \pi$, where $\|\cdot\|$ is the spectral norm. I'm trying to prove the following bounds on the Frobenius norm of the ...
Haimeng Zhao's user avatar
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1 answer
140 views

Renorming on a separable Banach space

Let us consider the sequence space $c_0$ with the equivalent norm $$\Vert x \Vert^2 = \max_{i\ge1} \vert x^i \vert^2 + \sum_{i=2}^{\infty} 2^{-i+1} \vert x^i \vert^2 $$ for $x=(x^1,x^2,\ldots)\in c_0$....
PPB's user avatar
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0 votes
1 answer
188 views

Norm functions induced by convex bodies

Given a centrally symmetric convex body $K$ in the plane (with smooth boundary), it is easy to see that there exists a norm function $g:\mathbb{R}^2\to \mathbb{R}_{\geq 0}$ for which $K$ is the unit ...
kvicente's user avatar
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2 votes
1 answer
107 views

prove with a probability of at least $1/e$: $\left\|\sum_{i=1}^k a_{i} P_{i}\right\|_2 \geq\left\|P_{1}\right\|_2$ holds

Let $a_i (i \in\{1...k\})$ be $k$ IID standard Gaussian random variables, $P_i$ are $d$-dimensional constant vectors. How to prove with a probability of at least $1/e$, $$ \left\|\sum_{i=1}^k a_{i} P_{...
Shuofeng Zhang's user avatar
0 votes
1 answer
130 views

Deducing norm concentration from MGF bounds

Suppose that $X$ is a centered, $\mathbf{R}^d$-valued random variable such that for all $t \in \mathbf{R}^d$, there holds the bound $$\log \mathbf{E} \left[ \exp \langle t, X \rangle \right] \leqslant ...
πr8's user avatar
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1 vote
1 answer
167 views

Norm of $2^{i}$-th primitive root

Let $ K $ be finite degree extension of $ \mathbb{Q} $ such that $ -1 $ is not a square in $ K $. Let $ L = \frac{K[x]}{\langle x^2 +1\rangle}$. Thus every element of $ L $ is of the form $ a + ib $ ...
Sky's user avatar
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0 votes
0 answers
113 views

Norm distance in a Banach space

Consider the Hilbert space $l_2(\mathbb{N})$ under the square summable norm $\Vert \cdot \Vert_2.$ Let us define a new norm $||| \cdot ||| $ equivalent to $\Vert \cdot \Vert_2$ such that the closed ...
PPB's user avatar
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2 votes
0 answers
52 views

Classification of submultiplicative ring norms on $\mathbb Q$

Let $R$ be a ring with identity. I call a non-negative real valued function $N: R \to \mathbb R_{\geq 0}$ a ring norm, if it has the following properties: $N(r) = 0$ iff $r = 0$ $N(r+s) \leq N(r) + N(...
Adelhart's user avatar
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1 answer
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Smoothness of a Hilbert space under an equivalent norm

Let us take the Hilbert space $l_2$ with an equivalent norm $\Vert x \Vert = \max \{2 \Vert x \Vert_1, \Vert x \Vert_2 \}$, where $\Vert x \Vert_1 =( \sum_{n=2}^\infty x_n^2 )^{\frac{1}{2}}$ and $\...
PPB's user avatar
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5 votes
0 answers
134 views

Is there a good notion of higher-rank archimedean norm?

Let $K$ be a field. I think I know what a norm (archimedean or not) $|-| : K \to \mathbb R_{\geq 0}$ is. In the case where the norm is nonarchimedean, it's equivalent to the data of a valuation of ...
Tim Campion's user avatar
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0 votes
1 answer
138 views

Finding weak LUR property of $C[0,1]$ with an equivalent norm

On the space $X=C[0,1]$, define a norm $||| f |||^2=\Vert f \Vert_{\infty}^2 + \Vert f \Vert_2^2$, where $\Vert \cdot \Vert_\infty$ is the sup norm on $C[0,1]$ space and $\Vert \cdot \Vert_2$ is the $...
PPB's user avatar
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1 vote
1 answer
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Norm inequality

In an article I read, I have the following inequality: $\|A-B\|_1 \geq \max \{ \|A 1_m- B 1_m \|_1, \|A^T 1_n - B^T1_n\|_1 \}$ Where $A, B \in \mathbb{R}_+^{m\times n}$. The $\|\cdot\|_1$ refers ...
CereIssou's user avatar
1 vote
0 answers
108 views

Distribution of norm over projected unit vectors

I am interested in the distribution of norms of projected unit vectors, for a particular class of projections. We first draw an $n$-dimensonal unit vector $v=X/||X||$ where $X=(X_1,X_2,\cdots, X_n)$ ...
galoistr93's user avatar
1 vote
1 answer
211 views

Third order matrix differential norm

Suppose we have a function $f:\mathbb{R}^n\to\mathbb{R}$ that is at least three times differentiable. Clearly, there is a relationship between the symmetric trilinear form $$T_1=\nabla^3f(x),$$ and ...
RS-Coop's user avatar
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3 votes
2 answers
234 views

Extend an inequality on matrix norms

Let $A$ denote an $n \times n$ matrix, and $\sigma_i(\cdot)$ denote $i$-th largest singular value. Can we extend the following result to general $p \geq 1$? For all $k = 1, \dots, n$, $$ \sum_{i = 1}^...
Xiangxiang Xu's user avatar
1 vote
0 answers
82 views

Name for natural norm on functions to non-linear targets

Is there a standard term for the quasi-norm $$\|f\|_{[k]}=\sum_{i=1}^k(\sup\|f^{(i)}\|)^{1/i}$$ ? It is useful due to the fact that it is reasonably compatible with post-composition by smooth ...
John Pardon's user avatar
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9 votes
1 answer
644 views

Banach space with uncountable basis

We know that an infinite dimensional Banach space has an uncountable Hamel basis. Now if $X$ is a vector space with an uncountable Hamel basis, does there exist a norm on $X$ for which $X$ is a Banach ...
Anupam's user avatar
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2 votes
1 answer
149 views

Is the measure density condition a necessary condition for bounding the Sobolev norm $W^{n,p}(\Omega)$ by the extremal terms?

Let $\Omega \subseteq \mathbb{R}^M$ be a measurable subset of positive measure. R. A. Adams and J. Fournier in their article have proven that if $\Omega$ satisfies the so-called weak cone property, ...
Kacper Kurowski's user avatar
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0 answers
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Probability of polynomials products to be bounded by a given bound

I am given a quotient ring $R=\frac{\mathbb{Z}[x]}{\left< x^n +t\right>}$ for $t\in\mathbb{Z}$, and two polynomials from $R$, $A$,$B$ and let $C$ to be there product. Defining the norms $$\Vert ...
Don Freecs's user avatar
2 votes
0 answers
86 views

Nuclear norm minimization of convolution matrix (circular matrix) with fast Fourier transform

I am reading a paper Recovery of Future Data via Convolution Nuclear Norm Minimization. Here, I know there is a definition for convolution matrix. Given any vector $\boldsymbol{x}=(x_1,x_2,\ldots,x_n)^...
Xinyu Chen's user avatar
5 votes
1 answer
245 views

Analytical form for the nuclear norm of an $n \times n$ matrix

I get the follow equation in a paper. Let $A \in \mathbb{R}^{2 \times 2}$, then $M = A^TA$ is a positive semi-definite matrix, the nuclear norm of $A$ is: $$ \Vert A \Vert_* = \sqrt{\operatorname{tr}(...
zhamao dra's user avatar
2 votes
0 answers
116 views

Norm of sections on $T$-invariant subspace of a homogenous space over $\mathbb{Q}_p$

Let $K=\mathbb{Q}_p$ and $G$ a split reductive group over $K$ with split maximal torus $T$. Furthermore, $X$ is a smooth projective variety over $K$ with a free $G$-action. Let $U \subset X$ be a $T$-...
KKD's user avatar
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3 votes
0 answers
134 views

Unitary equivalence of symmetric and homogenous polynomials

Given any two symmetric and homogenous polynomials with complex coefficients, I'm trying to determine if a unitary change of basis relates them. Specifically, assuming the polynomials are of degree $n$...
Deepesh Singh's user avatar
3 votes
2 answers
1k views

Is the matrix induced L1-norm greater than the induced L2-norm?

For $A \in \mathbb R^{m \times n}$ and the induced norms: $$ \| A \|_1 = \max_{x \ne 0} \frac{\|Ax\|_1}{\|x\|_1} $$ $$ \| A \|_2 = \max_{x \ne 0} \frac{\|Ax\|_2}{\|x\|_2} $$ ... where: $$ \|x\|_1 = \...
DrunkCoder's user avatar
2 votes
1 answer
97 views

Fix positive $t$. Construct $a_n \in \mathbb R^n$ such that $(\inf_x \|x-a_n\|_2 + t\|x\|_1 )/\min(\|a_n\|_2,t\|a_n\|_1) \to 0$

For any positive number $t$, nonnegative integer $n$, and nonzero vector $a \in \mathbb R^n$, define $$ \begin{split} K_n(a,t) &:= \inf_{x \in \mathbb R^n} \|x-a\|_2 + t\|x\|_1,\\ M_n(a,t) &:= ...
dohmatob's user avatar
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6 votes
0 answers
199 views

Mean value theorem for Dirichlet series of prime support?

Let $\{a_n\}_{1\leq n\geq N}$, $a_n\in \mathbb{C}$. Let $F(s) = \sum_{n=1}^N a_n n^{-s}$. By a mean-value theorem (Montgomery-Vaughan, 1973), $$\int_0^T |F(i t)|^2 = \sum_{n=1}^N |a_n|^2 (T + O(n)).$$ ...
H A Helfgott's user avatar
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0 votes
1 answer
135 views

How to prove this inequality for the norm $ \|\cdot\|_{1,\infty} $?

Let $ \{a_k\}_{i=1}^n $ is a positive sequence. For $ 0<p<\infty $, space $ L^{p,\infty} $ is defined by $$ \left\{f:\|f\|_{p,\infty}=\inf\left\{C>0:a_f(\lambda)\leq C/\lambda^p\right\}\right\...
Luis Yanka Annalisc's user avatar
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0 answers
86 views

Is there any way of explaining the Cayley/Beltrami–Klein metric to undergrads?

How to explain the Cayley-Klein or sometimes called Beltrami–Klein metric concept to find the distance between two points in a hyperbolic space to an audience with no higher education than maybe a ...
Dian Sheng's user avatar
1 vote
0 answers
75 views

Morphism in commutative square strict?

Let $G,H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism. Then $f$ is said to be strict if $G/\mathrm{Ker}(f) \cong \mathrm{Im}(f)$ is an isomorphism of topological ...
KKD's user avatar
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