# Questions tagged [norms]

The norms tag has no usage guidance.

252
questions

**2**

votes

**1**answer

75 views

### Estimate of Hölder Norms (Littlewood–Paley theory)

I'm currently studying Littlewood–Paley theory and its application to norm estimate/PDEs by reading Muscalu and Schlag's textbook, where I encountered the following norm estimate problem:
Recall that ...

**2**

votes

**0**answers

103 views

### Green's identity with a different norm

Let $\Omega \subset \mathbb{R}^n$ be a domain with a smooth boundary $\Gamma$. Suppose that $f, g \colon \mathbb{R}^n \to \mathbb{R}$ are of class $C^\infty( \overline{\Omega})$. Then Green's first ...

**2**

votes

**0**answers

94 views

### $L^p$ estimate of a multiplier operator

I'm studying harmonic analysis by myself and I encountered the following claim about multipliers: consider a sequence of complex numbers $\{m_{n}\}_{n \in \mathbb{Z}}$ that satisfies:
$$\sum_{n \in \...

**12**

votes

**3**answers

227 views

### Probability of $\ell_1$-norms of vertices of the rotated Hamming cube

Let $O$ be a $d$-dimensional rotation matrix (i.e., it has real entries and $OO^T = O^TO = I$). Let $\mathbf{x}$ be a uniformly random bitstring of length $d$, i.e., $\mathbf{x} \sim U(\{0,1\}^d)$. In ...

**-2**

votes

**1**answer

62 views

### A generalized norm function in $\mathbb{R}^n$ [closed]

We defined a new norm. The norm of $x \in \mathbb{R}^n$ is defined as
$$ N_P(x) = \min \{t \geq 0 : x \in t\cdot P\} \enspace,$$
where $P$ is a centrally symmetric and convex body centered at the ...

**1**

vote

**1**answer

36 views

### Double space integral formulation of homogeneous Sobolev norm

Define for $u\in C_c^\infty (\mathbb R^n), 0<s<1$ the integral
$$
I_s(u) = \int_{(x,y)\in \mathbb R^{n+n}} \frac{(u(x+y)-u(x))^2}{|y|^{d+2s}} dxdy.
$$
I wish to prove that for some $C=C(s)>1,$...

**0**

votes

**0**answers

73 views

### What is the standard terminology for the quantity $\|\nabla f\|_{L^2(\mu)} := \sqrt{\int_{\mathbb R^d}\|\nabla f(x)\|^2d\mu(x)}$?

Let $f:\mathbb R^d \to \mathbb R$ be a continuously differentiable function and let $\mu$ be a probability measure on $\mathbb R^d$.
Question. What is the standard teminology for the quantity $\|\...

**4**

votes

**1**answer

325 views

### Embedding of a Banach space into a Hilbert space

Let $\mathbb H$ be a Hilbert space and let $\mathbb B$ be a Banach space continuously embedded in $\mathbb H$ and distinct from $\mathbb H$. Is it true in general that $\mathbb B$ is an $F_\sigma$ of ...

**0**

votes

**1**answer

55 views

### Variance of spectral density is related to the gradient of signal?

Define the frequency variance as:
$$ \sigma^2 = \int^\infty_{-\infty}\omega^2 P(\omega) d\omega$$
Where $P(\omega)$ is the spectral density function, which is the same as normalized power. Therefore,
$...

**0**

votes

**0**answers

23 views

### Analytic formula for $D(x,y) := \sup_{z \in B_p} \|z-x\|_1 - \alpha\|z-y\|_1$, where $\alpha \ge 0$ and $B_p$ is the unit $L_p$-ball

Let $\alpha \in [0,\infty)$ and $p \in [1,\infty]$, and consider the function $D_\alpha:B_p \times B_p\to \mathbb R$ defined by
$$
D_{\alpha,p}(x,y) := \sup_{z \in B_p} \|z-x\|_1 - \alpha\|z-y\|_1,
$$...

**0**

votes

**0**answers

43 views

### Invariant matrix norm over integers

Let $A$ be an arbitrary matrix and $R$ a special matrix of the same order. We know that $\|RA\|=\|A\|$ when $R$ is unitary and $\|.\|$ is Frobenius/spectral norm. In case over integers, do we have ...

**1**

vote

**0**answers

47 views

### Traceless low rank approximation of a symmetric matrix by SVD

I have a symmetric matrix $M\in \mathcal{S}^n$ with rank $\mathbf{r}>2$. We can arrange its singular values by
$$(\sigma_1=|\lambda_1|)\geq (\sigma_2=|\lambda_2|)\geq \dots \geq (\sigma_r=|\...

**3**

votes

**1**answer

69 views

### Condition number for matrix of eigenvectors of a diagonalizable matrix

Let $A$ be a diagonalizable matrix, i.e., $A=SDS^{−1}$. For any matrix $A$, condition number is defined as $\kappa(A)=\|A\| \|A\|^{-1}$.
For $A$ being a diagonalizable matrix, define $G_A=\{{S: S^{-1} ...

**2**

votes

**0**answers

51 views

### Decomposition of the Orlicz norm into sequential norm

I am bearing seeking for a sequential decomposition of the norm in Orlicz space.
Let me state what is known in the particular case of Lebesgue space $L^p(\Bbb R^d)$.
Given $u\in L^p(\Bbb R^d)$ let
$$n\...

**0**

votes

**1**answer

60 views

### Weighted metric (or semi-metric) with incorporated distance between Dimensions

Im trying to construct a distance Measure between two vectors, that takes into account also the distance between the Dimensions. I will illustrate my Problem with some examples:
$x,y \in \mathbb{R}^n$,...

**4**

votes

**1**answer

93 views

### Uniform smoothness inequality for Schatten norms

I've previously asked this question on stack exchange.
I'm looking for a proof of the inequality
$$
\left[ \frac12(\left\|A+B\right\|_p^p + \left\|A-B\right\|_p^p)\right]^{2/p} \leq \left\|A\right\|_p^...

**3**

votes

**1**answer

50 views

### 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

88 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\...

**7**

votes

**2**answers

447 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 ...

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votes

**0**answers

50 views

### Double dual of an extended seminorm

I have already ask this question on MSE but it didn't received any answer during the weekend. So I thought I will ask it here too. Please let me know if it is bad practice and/or which of MO and MSE ...

**11**

votes

**1**answer

364 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
$$ \limsup \|x_n - y\| < \limsup \|x_n\| ?$$
...

**2**

votes

**2**answers

114 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\...

**1**

vote

**0**answers

44 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 ...

**2**

votes

**0**answers

105 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 ...

**4**

votes

**0**answers

121 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 ...

**2**

votes

**1**answer

173 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

58 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

296 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 \...

**2**

votes

**0**answers

108 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_{\...

**0**

votes

**0**answers

76 views

### Generalization of Banach-Lamperti theorem to element-wise nonlinear transformations

According to the Banach-Lamperti theorem, every linear isometry $T$ of $\ell_p = \ell_p(\mathbb{N})$ (with $1 \leq p < \infty,𝑝\ne 2$) is of the form $T:(a_n)↦(\epsilon(n)a_{\sigma(n)})$, where $\...

**5**

votes

**1**answer

207 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

994 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-...

**2**

votes

**0**answers

45 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....

**0**

votes

**1**answer

110 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 ...

**3**

votes

**1**answer

240 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 ...

**9**

votes

**0**answers

235 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 ...

**3**

votes

**2**answers

210 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 $|\cdot|_{prod}$...

**-1**

votes

**2**answers

205 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 $X$?

**2**

votes

**1**answer

258 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 ...

**1**

vote

**1**answer

162 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

204 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

126 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

494 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

121 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

77 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

191 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

268 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

148 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

79 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|\...

**7**

votes

**1**answer

606 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 ...