Questions tagged [norms]

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Estimate of the norm of the radial part of a function

Consider a function $u\in L^2(\mathbb R^N)$, and another function $\varphi$ which is the unique solution to the Poisson equation $\Delta \varphi = u$ vanishing at $\infty.$ We know that the radial ...
Ma Joad's user avatar
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1 vote
1 answer
131 views

Norm of a matrix with clustered eigenvalues

On page 271 of Trefethen and Bau's Numerical Linear Algebra, it is constructed a matrix $$A=2I_{m\times m}+0.5\cdot\frac{\text{rand}(m)}{\sqrt{m}}$$ for $m=200$, where rand(m) is an array with $m\...
Leibniz's user avatar
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8 votes
3 answers
2k views

Bounding supremum norm of Lipschitz function by L1 norm

Consider $f:[0,1]^d \to \mathbb{R}$. Suppose that $f$ is $L$-Lipschitz w.r.t. the Euclidean norm. Can we provide an upper bound on $\|f\|_\infty$ in terms of $\|f\|_1 := \int_{[0,1]^d} |f(x)|dx$ ? In ...
Aurelien's user avatar
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12 votes
1 answer
441 views

Subtracting the weak limit reduces the norm in the limit

Question Let $X$ be some reflexive Banach space. Suppose $x_n$ is some sequence in $X$ that weak converges to some $y \neq 0$. Is it the case that $$ \limsup \|x_n - y\| < \limsup \|x_n\| ?$$ ...
Willie Wong's user avatar
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2 votes
2 answers
257 views

Minimum Euclidean squared norm in the convex hull of points with rational coordinates

This is probably known, but I have not located a reference. Let $P$ be the convex hull of $k$ points in $\mathbb R^n$ with rational coordinates. Consider the Euclidean square norm function $F:P\to\...
Claudio Gorodski's user avatar
2 votes
1 answer
248 views

Is the product of two Banach algebras given by the injective cross-norm itself a Banach algebra?

I understand that you can take the tensor product of Banach spaces in many different ways by specifying different norms; of particular interest to me are the cross-norms. The projective and injective ...
FeralX's user avatar
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2 votes
0 answers
127 views

Are there any known algebras or vector spaces, where absolute value, modulus or norm is connected to the factors of $\pi$ or $e^{-\gamma}$?

I am currently working on an algebra of divergent integrals and series, and all the elements of that space consist of a regular part (which is a real or complex number) and irregular part (which is ...
Anixx's user avatar
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4 votes
0 answers
165 views

Condition on kernel convolution operator

I am studying O'Neil's convolution inequality. Let $\Phi_1$ and $\Phi_2$ be $N$-functions, with $$ \Phi_i(2t)\approx \Phi_i(t), \quad i=1,2 $$ with $t\gg 1$ and let $k \in M_+(\mathbf R^n)$ is the ...
Forbs's user avatar
  • 101
2 votes
1 answer
414 views

How to find upper and lower bound

Let $\Sigma \in S_{++}^n$ be a symmetric positive definite matrix with all diagonal entries equal to one. Let $U \in \mathbb{R}^{n \times k_1}$, $W \in \mathbb{R}^{n \times k_2}$, $\Lambda \in \mathbb{...
newbie's user avatar
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0 answers
167 views

Ratio of maximum to minimum value

Let $y = X \beta + \epsilon$, where $y \in R^{n}$, $X \in R^{n \times p}$, $\beta \in R^{p}$ and $\epsilon \in R^{n}$. Let $X = USV^\top$ be the SVD of the $X$. Let $u_i$ be the rows of $U$, then ...
newbie's user avatar
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1 vote
1 answer
2k views

How to minimize l1-norm constrained by "infinity norm"

Let $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m $. I have the following two problems: P.1. \begin{equation} \underset{x\in\mathbb{R}^n}{\text{minimize}} \| Ax-b \|_1 \\ \text{s.t. } \| x \...
Paul Goyes's user avatar
2 votes
0 answers
129 views

Conditions on the inequality with a gauge norm

Let $\Phi(x)=\int_0^x \phi(y)\,dy$, $x \in \mathbb{R}_+$, be an N-function, and let $u$ be locally inferable on $\mathbb{R}_+$. Consider the gauge norm $$ \rho_{\Phi,u}(f)=\inf\{\lambda>0: \int_{\...
user124297's user avatar
6 votes
1 answer
274 views

Recover approximate monotonicity of induced norms

Let $A$ some square matrix with real entries. Take any norm $\|\cdot\|$ consistent with a vector norm. Gelfand's formula tells us that $\rho(A) = \lim_{n \rightarrow \infty} \|A^n\|^{1/n}$. Moreover, ...
ippiki-ookami's user avatar
15 votes
2 answers
2k views

In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?

This is a cross-posted on MSE here. Let $(X,d)$ be a metric space. Say that $x_n\in X$ is a P-sequence if $\lim_{n\rightarrow\infty}d(x_n,y)$ converges for every $y\in X.$ Say that $(X,d)$ is P-...
Nikhil Sahoo's user avatar
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2 votes
0 answers
164 views

Removing integral from norm by inequality

My first question on Math Overflow. For my Mathematics Bachelor thesis I am looking at a paper called "Deep Limits of Residual Neural networks" by Matthew Thorpe and Yves van Gennip. (arxiv....
The Coding Wombat's user avatar
0 votes
1 answer
655 views

Prove the optimal solution to maximizing nuclear norm with constraints is attained at corner points of feasible region

The nuclear norm (trace norm) of a matrix $X \in \Bbb R^{m \times n}$ is defined as $$\|X\|_* := \sum_{i=1}^{\min(m,n)} \sigma_i(X)$$ where $\sigma_i(X)$ are the singular values of $X$. The ...
Jack's user avatar
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3 votes
1 answer
324 views

Seminorm which is zero on dense subset

Let $X$ be a Banach space and let $\hat{X}$ be a dense subset of $X$. If $p$ is a seminorm on $X$ such that $p(x) =0 $ for all $x \in \hat{X}$, does $p(x) =0$ for all $x\in X$ (is $p$ the trivial ...
Shash's user avatar
  • 41
10 votes
0 answers
253 views

Integral points on elliptic curve and the Lee norm

This question is based on small experiments I have done in Sagemath and if it is not research level, I will move it to MSE: Let $E$ be an elliptic curve defined with coefficients in $\mathbb{Z}$. The ...
user avatar
3 votes
2 answers
374 views

Norm on tensor product of fields

Let $F$ be an algebraically closed field of characteristic $p$ equipped with an absolute value $|\cdot|:F \rightarrow \mathbb{R}_{\ge 0}$ with respect to which $F$ is complete. Define $|\cdot|_{prod}$...
user223794's user avatar
-1 votes
2 answers
468 views

inequivalent norms [closed]

I am thinking about the following question: Let $X$ be a Banach space, say separable, e.g., $l_p$ or $c_0$. When can I say that there exist inequivalent complete norms on $X$?
user92646's user avatar
  • 617
2 votes
1 answer
502 views

Separable Banach spaces isometric to quotient of a Banach space

We know that every separable Banach space is isometrically isomorphic to a quotient space of $(\ell^1,\|.\|_1)$. We also know that the norm defined by $\|x\|=(\|x\|_1^2+\|x\|_2^2)^{1/2}$ for all $x\in ...
Anupam's user avatar
  • 479
1 vote
1 answer
207 views

Does kernel regression preserve monotonicity?

Consider the Kernel regression estimator: $$\hat{y}(x)=\frac{\sum_{i=1}^n{K(x-x_i)y_i}}{\sum_{i=1}^n{K(x-x_i)}},$$ where $x,x_1,\dots,x_n\in\mathbb{R}^d$, $y_1,\dots,y_n\in\mathbb{R}$, where $K:\...
cfp's user avatar
  • 183
2 votes
1 answer
407 views

Conditions such that norm of matrix vector can be written as the derivative of the norm of the vector for some convex fonction

Problem statement: Let $A$ be a matrix $\mathbb{R}^{d \times d}$, I want to find some conditions on $A$ such that there exists a differentiable convex function $f: \mathbb{R_{+}} \rightarrow \mathbb{...
Titouan Vayer's user avatar
2 votes
0 answers
260 views

Is this an error in Loomis and Sternberg?

In Loomis and Sternberg's Advanced Calculus section 3.3 Continuity, they make this comment (just before Theorem 3.2, [pp. 128-129 in my copy): A linear map $T : V \rightarrow W$ is bounded below by ...
Robin Adams's user avatar
6 votes
1 answer
978 views

Cartesian product of Banach spaces: all norms such that the inclusion is an isometry are equivalent?

Let $\mathcal{A}$ be an arbitrary (typically infinite-dimensional) Banach space with norm $\|\cdot\|_{\mathcal{A}}$ and let $\mathcal{A}^{n}$ be its Cartesian product. I came across the following ...
Peter's user avatar
  • 131
0 votes
1 answer
301 views

$L^p$ norm inequalities with respect to strongly-log-concave densities

Let $\pi(x)=\frac{e^{-f(x)}}{\int_{\mathbb{R}^d}e^{-f(u)}du}$ be a strongly-log-concave distribution, i.e., $f(x):\mathbb{R}^d\rightarrow R$ is an $m$-strongly convex function. Also, $f(x)$ has $L$-...
Sayan's user avatar
  • 9
2 votes
1 answer
117 views

When does the map from a normed vector cone to its double dual preserve norms?

If $V$ is a normed vector space then the natural map from $V$ to its double dual $V''$ is norm-preserving as follows from Hahn-Banach theorem. This is well-known. Now assume that P is just a vector ...
Sergey Slavnov's user avatar
2 votes
1 answer
351 views

Is the union of l^p a Banach space under some norm?

As a set of sequences, take the union of $\ell^p$, $p\geq 1$. As $p$ increases, the $\ell ^p$ space is larger, with strict inclusion. However, this infinite union is strictly contained in $c_0$, ...
Pow's user avatar
  • 31
4 votes
0 answers
479 views

analytic approximations of the min and max operators

Question: What is the state of the art on analytic approximations of $\min$ and $\max$? My hunch is that numerical analysts probably have a better solution than the one I propose here. For any $\...
Aidan Rocke's user avatar
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0 votes
0 answers
317 views

Spectral norm of difference of quadratic matrices restricted to a subspace

Say that we have two matrices $X$ and $Y$ of dimensions $(T \times N)$ with $N < T$ and $rank(X)=rank(Y)=N$. Furthermore, define a $(T \times k)$ dimensional matrix $D$ with $k<N$ and $rank(D)=k$...
E_Wijler's user avatar
-1 votes
1 answer
317 views

Sequence converging to different limits with respect to two different _complete_ norms

Do there exist a real vector space $X$ complete with respect to norms $|\cdot|$ and $\|\cdot\|$ and a sequence $(x_n)_{n\in \mathbb N} \subset X$ such that there exist $x,y\in X$: $x\ne y$, $|x_n - x|\...
Skeeve's user avatar
  • 1,277
7 votes
1 answer
1k views

Operator norm of square root of matrix vs original

If I have a nonsymmetric matrix whose operator norm is $\leq 1$ and square root it, does its operator norm remain below $1$? More formally, I want to know whether there is always at least one square ...
user149538's user avatar
1 vote
0 answers
59 views

What is the distance of a particular root to the farthest one with respect to it as a function of a compacting factor?

Let $f:\mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function defined as $$f := 2a_{1}(x - x^{p}) + 2a_{2}(x - x^{p})\sum_{k \in K}||x_{k} - x^{p}_{k}||^{2} + \dfrac{2a_{3}}{\alpha}\sum_{j \in J}\dfrac{...
User's user avatar
  • 206
-1 votes
1 answer
284 views

Holder inequality for a general rectangular matrix

Let $A \in \mathbb{R}^{m\times n}$ and $p,q \in \mathbb{R}^{+}$ such that $\frac{1}{p}+\frac{1}{q}=1$. I am interested to prove the following: $$ \|A\|_{p}=\|A^T\|_q$$ I have tried using Holder ...
Prashant Govindarajan's user avatar
0 votes
0 answers
97 views

What is the closed-form solution to this double-sum norm function?

Given two points $A,B \in \mathbb{R}^2$, one defines the Euclidean distance $f: \mathbb{R}^2\times\mathbb{R}^2 \rightarrow \mathbb{R}^{\ge 0}$ as follows. $$f(A,B) := \Vert A-B \Vert : = \sqrt{(A_{x}...
Pinton's user avatar
  • 109
3 votes
0 answers
131 views

Lower bound on the intersection of $\ell_1$ $n$-balls

Let $B_1$ and $B_2$ be two balls in $\mathbb{R}^n$ in $\ell_1$ norm, with distance $d$ and radius $R$. Is there a lower bound on the volume of the intersection between the two n-balls? (assuming the ...
GWB's user avatar
  • 181
5 votes
1 answer
242 views

What does the image of the integer lattice under a norm look like?

The question that I shall ask here has arisen in the context of Diophantine approximation. I find it rather interesting, and I have no idea how to answer it. Any help, advice, or suggestions for ...
Mishel Skenderi's user avatar
3 votes
0 answers
223 views

Generalization of Ostrowski's Theorem

Let $\Lambda$ be a totally ordered set with two binary operations $+$ and $\cdot$ on it, such that: 1) $+$ is associative, has a unit $0$, and is symmetric. 2) $\cdot$ is associative, has a unit $1$,...
Ronald J. Zallman's user avatar
2 votes
1 answer
278 views

A problem in Bushnell and Henniart's book, "The local Langlands conjecture for GL(2)"

On page 123 of Chapter 5 in Bushnell and Henniart's book The Local Langlands Conjecture for GL(2), they state an elementary property of tamely ramified extension of local fields, which is as follows, ...
Qingzhi Li's user avatar
2 votes
1 answer
3k views

How to calculate or estimate RKHS norm? [closed]

I am working with GP-UCB and need to calculate RKHS norm as in Theorem 6 of Srinivas et.al 2012. I found on page 3 column 1 like: The induced RKHS norm $||{f}||_k=\sqrt{<f,f>}_k$ measures ...
Parikshit's user avatar
1 vote
1 answer
267 views

Density of norm-attaining operators

By Bishop-Phelps theorem we know that for a real Banach space, the set of all norm attaining bounded linear functionals is norm-dense in $X^*$, the topological dual of $X$. We also know that in ...
Anupam's user avatar
  • 479
14 votes
1 answer
835 views

What are the applications of the Mazur-Ulam Theorem?

Every bijective isometry between normed spaces is affine. This well-known and beautiful statement, the Mazur-Ulam Theorem, was proved in 1932, but the proof has been simplified and polished in years, ...
Pietro Majer's user avatar
  • 56.5k
3 votes
0 answers
102 views

"Hoelder conjugate" version of the Johnson-Lindenstrauss transform

A variation of the well-known Johnson-Lindenstrauss transform (JLT) asserts that for $x_1,\ldots,x_m\in\mathbb{R}^n$ there exists a linear transformation $A:\mathbb{R}^n\to\mathbb{R}^k$ with $k=\...
user134977's user avatar
5 votes
1 answer
191 views

Expected supremum of normalised random walk

Let $X^i\in \mathbb R^d$ be iid. random variables for $i=1$ to $n$. Assume $\mathbb E[X^i]=0$ and the covariance matrix $\mathbb C[X^i] = \mathbb E[X^iX^{iT}] = I$ is the identity matrix. Define $S^k=...
Thomas Dybdahl Ahle's user avatar
7 votes
3 answers
897 views

Norms as Points in $C(X)$

$\newcommand\abs[1]{\lvert{#1}\rvert}$Let $X$ be a compact hausdorff space, and put $C(X)$ for the $\mathbb{R}$-algebra of continuous maps from $X$ to $\mathbb{R}$. For each point $x$, there is a ...
Ronald J. Zallman's user avatar
1 vote
1 answer
122 views

Matrix inequalities for the moment of the fixed Shatten norm

Let $A_i, i=1, \ldots, N$ be real (or complex) matrices of the same dimension. Let $r_i, i=1, \ldots, N$ be independent Rademacher random variables. The following inequality gives a bound on the ...
user124297's user avatar
4 votes
0 answers
121 views

$L^1$ norm of oscillatory integral operator

My question is about the $L^1_x$ norm of an oscillatory integral like $$ \int_{\mathbb{R}^n} e^{i(y\cdot x+\lambda \phi(y))}f(y)dy,$$ where $\lambda \in \mathbb{R}$, $f\in C^{\infty}_c(\mathbb{R}^n)$ ...
Paul's user avatar
  • 141
2 votes
0 answers
30 views

Find the point that minimizes the summation of L_\infty norms to three given points

Given three points $\omega_1$, $\omega_2$, $\omega_3 \in \mathbb{R}^d$, how can I find the point $\omega \in \mathbb{R}^d$ such that the summation of its $\ell_\infty$ distances to these three points ...
liyan's user avatar
  • 33
0 votes
1 answer
57 views

Upper-bounds for a vector equation

Let $a$, $c$, $d$, $v \in \mathbb{R}^n$ are vectors, and $A, B \in \mathbb{R}^{n \times n}$ are matrices. Suppose that $ v = Ac-d $, and $a = ABc- \|B\| d$ where $\| B \|$ is the maximum value of the ...
livehhh's user avatar
  • 85
6 votes
1 answer
246 views

Area of $n$-sphere contained outside $\ell_1$ ball

For a given $r>1$, what is the surface area of $\mathbb S^{n-1}$ (the sphere of radius 1 in $\mathbb R^n$) which is contained outside of the $\ell_1$ ball of radius $r$? Or equivalently, if $X\sim ...
Shant Boodaghians's user avatar

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