Questions tagged [norms]
The norms tag has no usage guidance.
334
questions
3
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70
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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 ...
1
vote
1
answer
131
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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\...
8
votes
3
answers
2k
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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 ...
12
votes
1
answer
441
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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
$$ \limsup \|x_n - y\| < \limsup \|x_n\| ?$$
...
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\...
2
votes
1
answer
248
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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 ...
2
votes
0
answers
127
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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 ...
4
votes
0
answers
165
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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 ...
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{...
0
votes
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 ...
1
vote
1
answer
2k
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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 \...
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
$$
\rho_{\Phi,u}(f)=\inf\{\lambda>0: \int_{\...
6
votes
1
answer
274
views
Recover approximate monotonicity of induced norms
Let $A$ some square matrix with real entries.
Take any norm $\|\cdot\|$ consistent with a vector norm.
Gelfand's formula tells us that $\rho(A) = \lim_{n \rightarrow \infty} \|A^n\|^{1/n}$.
Moreover, ...
15
votes
2
answers
2k
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In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?
This is a cross-posted on MSE here.
Let $(X,d)$ be a metric space. Say that $x_n\in X$ is a P-sequence if $\lim_{n\rightarrow\infty}d(x_n,y)$ converges for every $y\in X.$ Say that $(X,d)$ is P-...
2
votes
0
answers
164
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Removing integral from norm by inequality
My first question on Math Overflow.
For my Mathematics Bachelor thesis I am looking at a paper called "Deep Limits of Residual Neural networks" by Matthew Thorpe and Yves van Gennip. (arxiv....
0
votes
1
answer
655
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Prove the optimal solution to maximizing nuclear norm with constraints is attained at corner points of feasible region
The nuclear norm (trace norm) of a matrix $X \in \Bbb R^{m \times n}$ is defined as
$$\|X\|_* := \sum_{i=1}^{\min(m,n)} \sigma_i(X)$$
where $\sigma_i(X)$ are the singular values of $X$.
The ...
3
votes
1
answer
324
views
Seminorm which is zero on dense subset
Let $X$ be a Banach space and let $\hat{X}$ be a dense subset of $X$. If $p$ is a seminorm on $X$ such that $p(x) =0 $ for all $x \in \hat{X}$, does $p(x) =0$ for all $x\in X$ (is $p$ the trivial ...
10
votes
0
answers
253
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Integral points on elliptic curve and the Lee norm
This question is based on small experiments I have done in Sagemath and if it is not research level, I will move it to MSE:
Let $E$ be an elliptic curve defined with coefficients in $\mathbb{Z}$.
The ...
3
votes
2
answers
374
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Norm on tensor product of fields
Let $F$ be an algebraically closed field of characteristic $p$ equipped with an absolute value $|\cdot|:F \rightarrow \mathbb{R}_{\ge 0}$ with respect to which $F$ is complete.
Define $|\cdot|_{prod}$...
-1
votes
2
answers
468
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inequivalent norms [closed]
I am thinking about the following question:
Let $X$ be a Banach space, say separable, e.g., $l_p$ or $c_0$.
When can I say that there exist inequivalent complete norms on $X$?
2
votes
1
answer
502
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Separable Banach spaces isometric to quotient of a Banach space
We know that every separable Banach space is isometrically isomorphic to a quotient space of $(\ell^1,\|.\|_1)$. We also know that the norm defined by $\|x\|=(\|x\|_1^2+\|x\|_2^2)^{1/2}$ for all $x\in ...
1
vote
1
answer
207
views
Does kernel regression preserve monotonicity?
Consider the Kernel regression estimator:
$$\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:\...
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{...
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 ...
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 ...
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$-...
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 ...
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$, ...
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 $\...
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$...
-1
votes
1
answer
317
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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|\...
7
votes
1
answer
1k
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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 ...
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{...
-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 ...
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}...
3
votes
0
answers
131
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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 ...
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 ...
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$,...
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,
...
2
votes
1
answer
3k
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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 ...
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 ...
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, ...
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=\...
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=...
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 ...
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 ...
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 $\lambda \in \mathbb{R}$, $f\in C^{\infty}_c(\mathbb{R}^n)$ ...
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 ...
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 ...
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 ...