Questions tagged [norms]

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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 ...
Priyanka Priyadarshini Behera's user avatar
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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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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 $\...
Priyanka Priyadarshini Behera's user avatar
4 votes
0 answers
103 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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1 answer
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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 $...
Priyanka Priyadarshini Behera's user avatar
1 vote
1 answer
72 views

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
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74 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
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37 views

A Pinching inequality with respect to a continuous basis

For a generic trace-class matrix $A=\sum_{n,m}A_{n,m}|n\rangle\langle m|$, one can easily find the bound $\|P[A]\|_1\leq\|A\|_1\leq \|A\|_{1,1}$, where $\|A\|_1=\sum_{k}\sigma_k(A)$ is the trace norm, ...
La buba's user avatar
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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
221 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
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0 answers
69 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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Recovery from $p$-norms: Question on proof

I am reading a paper on recovery from power sums. A main statement of the paper is that for a vector $(a_1,...,a_n) \in \mathbb{N}^n$ with pairwise distinct $a_i$ and the map $$\phi:\begin{cases} ...
Qualearn's user avatar
9 votes
1 answer
359 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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1 answer
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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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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
69 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
197 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
113 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
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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
677 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
90 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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0 answers
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Expectation of a norm in Monte-Carlo method

I am currently studying the Monte Carlo methods for solving PDEs with random coefficients. My problem here is basically just doing with some algebraic properties of the expected value function which I ...
user492649's user avatar
6 votes
0 answers
169 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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1 answer
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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
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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
68 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
  • 451
4 votes
2 answers
226 views

A space isometric to $\ell_\infty^2$

Consider a norm on $\mathbb C^2$ as $\|(z_1,z_2)\|:=\max\{|z_1|,|z_2|,\frac{1}{\sqrt{2}}|z_1+iz_2|\}.$ Question. Is $(\mathbb C^2,\|.\|)$ linearly isometric to $(\mathbb C^2,\|.\|_{\infty})$ where $\|(...
A beginner mathmatician's user avatar
4 votes
1 answer
179 views

Is a mixture of $\ell_p$-norms $\eta(x):=\lVert x\rVert_2 + r\lVert x\rVert_p$ always dimensionlessly equivalent to some $\ell_q$-norm?

$\newcommand\norm[1]{\lVert#1\rVert}$For any $p \in [1,2]$, $r \ge 0$, and integer $d \ge 1$, define a mixed-norm $\eta:\mathbb R^d \to \mathbb R$ by $\eta(x) := \norm x_2 + r\norm x_p$, for any $x \...
dohmatob's user avatar
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On comparing the nuclear norm with the Hilbert-Schmidt norm for symmetric tensors

I am interested in the special case of a symmetric tensor $T_{i_1,\ldots,i_k}$ of rank $k$, where each index, say $i_\kappa$, where $1 \leq \kappa \leq k$, runs from $1$ to $2$. The entries of such a ...
Malkoun's user avatar
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2 votes
0 answers
71 views

Comparing the truncated $\ell^{1}$-norm of polynomial coefficients with the supremum norm on the unit disc

Let $p=a_{0}+a_{1}z+\ldots+a_{n}z^{n}$ be a polynomial. Consider the following truncated $\ell^{1}$-seminorm of the coefficients of $p$: $$\|p\|_{\ell^{1},\text{trun.}}:=\sum_{k=1}^{n}|a_{k}|=\|p-a_{0}...
Calculix's user avatar
  • 321
2 votes
2 answers
155 views

Prove spectral equivalence of matrices

Let $A,D \in \mathbb{R}^{n\times n}$ be two positive definite matrices given by $$ D = \begin{bmatrix} 1 & -1 & 0 & 0 & \dots & 0\\ -1 & 2 & -1 & 0 & \dots & 0\\...
Luna947's user avatar
  • 31
3 votes
1 answer
48 views

When do Orlicz norms tend to the uniform norm?

It is well known that the $p$-norms tend to the $\infty$-norm, in that if $\lVert f \rVert_q < \infty$ for some $q \ge 1$ then $\lVert f \rVert_p \to \lVert f \rVert_\infty$ as $p \to \infty$. ...
Olius's user avatar
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2 votes
0 answers
34 views

Example when Lorentz-Shimogaki condition satisfied with a specific Young's function

Let $\Psi(t)=\int_0^t\psi(s)$ be a Young's function. Then, $\Psi$ satisfies the Lorentz-Shimogaki condition if $$ \int_0^{\infty}\frac{\Psi(st)}{v(t)^2}\psi(t)dt< \infty. $$ Denote $\rho_{\Psi}=\...
user124297's user avatar
3 votes
0 answers
70 views

Example of the bounded convolution operator when Sharpley's conditions does not hold

I am reading about Orlicz and Marcinkevich spaces, and wondering whether there is an example in which Sharpley's condition is not satisfied for a special bounded operator $T_k$ (see for reference ...
volond's user avatar
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1 vote
0 answers
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If two functions are close apart can I proof the difference of their empirical loss is also small?

I am trying to understand the proof of Theorem 3 in the paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. Basically there exist atleast one $w_{L,e}$ in $\...
user avatar
3 votes
0 answers
100 views

Besov space norms

We need to recall some Besov space norms to formulate the question. Let $d \in \mathbb N$, $0<s<2, 1 \le p,q \le \infty$. Then the Besov space $B^s_{p,q}(\mathbb R^d)$ is given by the norm $$ \...
Paul Pfeiffer's user avatar
2 votes
1 answer
176 views

An upper bound on an invertible matrix

I have looked through books such as Matrix Analysis by R.A. Horn and C.R. Johnson and would not find an answer to the following question: Given $V^TV \in S^{n}$, where $V$ is an invertible matrix with ...
muddy's user avatar
  • 31
1 vote
2 answers
145 views

When is a natural map between completions injective?

Let $X$ be a vector space equipped with a norm $p$ and a seminorm $q$. Denote the completion of $X$ with respect to $p$ with $X_p$ and with respect to $p+q$ by $X_{p+q}$. Then the induced map $\iota : ...
iolo's user avatar
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0 answers
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Existence of distance-preserving mappings for general norm in vector space

We say a mapping $f:\mathbb R^n\to \mathbb R^n$ be 1-Lipschitz with respect to a norm $\|\cdot\|$ if $\|f(x)-f(z)\|\le\|x-z\|$ holds for all $x,z\in\mathbb R^n$. Such a mapping are sometimes called a ...
zbh2047's user avatar
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4 votes
0 answers
237 views

Estimates of the Frobenius norm of commutator

Let $A,B$ be two unitary matrices in $U(n)$, and $\|\cdot\|_{F}$ denote the Frobenius norm (or Hilbert Schmidt norm on the finite dimensional $M_n(\mathbb{C})$). I am looking for estimates of the ...
BharatRam's user avatar
  • 939
-1 votes
1 answer
70 views

"Large" compact sets in separable normed space

Let $(X, \lVert \cdot \rVert)$ be a separable normed space. Can we always guarantee that there is a nonempty compact set $K \subseteq B_X$, where $B_X$ is a closed unit ball in $X$ such that: $$\...
Kacper Kurowski's user avatar
0 votes
0 answers
141 views

Is there a relationship between infinity norm (or any other norms) of a vector and the trace of its covariance matrix?

I wish to know if there is a known relationship between the infinity norm (or any other norms) of a vector and the trace of its covariance matrix? I have found a paper that used the following ...
Spring Breeze's user avatar
1 vote
0 answers
92 views

Regularity of functions everywhere approximable by $n$-th degree polynomials

Let $(X, \lVert \cdot \rVert_X)$, $(Y, \lVert \cdot \rVert_Y)$ be two Banach spaces. A function $P \colon X \to Y$ such that there exists $n \in \mathbb{N}$ such that for all $i \in \{ 0, \ldots, n \}$...
Kacper Kurowski's user avatar
2 votes
1 answer
101 views

Can anything be said about the roots of the L4 center?

Modes, Medians and Means: A Unifying Perspective defines the following centers based on the $L_p$ norms: $$ \begin{aligned} \text{mode of x} = \arg \min_s \sum_i \lvert x_i - s \rvert^0 \\ \text{...
user19087's user avatar
  • 123
2 votes
0 answers
91 views

A truncated Frobenius norm of a matrix is convex or not?

Given a positive integer $k$ and a matrix $X\in \mathbb{R}^{m\times n}$. A truncated frobenius norm of a matrix $X$ is defined by $$\Vert X \Vert_{k,F} = \sqrt{\sum_{i=k+1}^{m} \sigma_i^2(X)},$$ where ...
Tung's user avatar
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3 votes
0 answers
144 views

Hausdorff measure of the unit ball of a norm on $\mathbb{R}^n$ is a universal constant

In [1], Kirchheim proved the area formula for Lipschitz maps $f\colon \mathbb{R}^n\to X$ where $X$ is an arbitrary metric space, using the notion of metric differentiability. The metric derivative of $...
Behnam Esmayli's user avatar
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0 answers
52 views

Existence of minimal subset of dual ball such that the intersection of kernels is trivial

Let $(V, \lVert \cdot \rVert)$ be a separable Banach space and let $B_{V^*}$ denote the closed ball in the dual $V^*$. Suppose we have a family $C \subseteq B_{V^*}$ such that $\bigcap_{\Lambda \in C} ...
Kacper Kurowski's user avatar
4 votes
1 answer
112 views

Is $0$ a member of the following special kind of a convex compact set?

Let $(V, \lVert \cdot \rVert)$ be a normed space. Let us consider the set $C = [-1,1]^{\dim V}$. The boundary of this set consists of closed subsets $B_i$ (indexed by some set $I$) of affine ...
Kacper Kurowski's user avatar
5 votes
2 answers
178 views

Which convex subsets of a normed space are intersections of balls?

Let $(V, \lVert \cdot \rVert)$ be a normed space. For any $A \subseteq V$, let $O(A)$ be the intersection of all closed balls containing $A$, or more precisely, let $O \colon 2^V \to 2^V$ be defined ...
Kacper Kurowski's user avatar
2 votes
1 answer
116 views

Relaxations for the spectral norm maximization problem

Optimizing the spectral norm of some positive semidefinite matrix $A(x) \in S^{n}$, w.r.t. a list of variables $x \in \mathbb{R}^d$ and semidefinite constraints is, in general, a nonconvex problem (...
ccln's user avatar
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