# Questions tagged [norms]

The norms tag has no usage guidance.

311
questions

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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 ...

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41
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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(...

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1
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97
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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 $\...

4
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103
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### 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 ...

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1
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87
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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 $...

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1
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72
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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 ...

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74
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### 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 $X=(X_1,X_2,\cdots, X_n)$ ...

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### 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, ...

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1
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107
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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 ...

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221
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### 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}^...

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0
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69
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### 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 ...

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39
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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}
...

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1
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359
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### 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 ...

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98
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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, ...

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51
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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 ...

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0
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69
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### 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)^...

5
votes

1
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197
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### 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}(...

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113
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### 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 $T$-...

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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 $n$...

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677
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### 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 = \...

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votes

1
answer

90
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### 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) &:= ...

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51
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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 ...

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169
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### 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),
$$\int_0^T |F(i t)|^2 = \sum_{n=1}^N |a_n|^2 (T + O(n)).$$
...

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1
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123
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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\...

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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 ...

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68
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### 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 ...

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votes

2
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226
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### 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 $\|(...

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### 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 \...

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41
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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 ...

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### 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}...

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votes

2
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155
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### 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\\...

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votes

1
answer

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### 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 $p \to \infty$. ...

2
votes

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answers

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### 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}=\...

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### 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 ...

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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 $\...

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### 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
$$ \...

2
votes

1
answer

176
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### 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 ...

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2
answers

145
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### 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 : ...

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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 ...

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### 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 ...

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votes

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### "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:
$$\...

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141
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### 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 ...

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92
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### 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 $i \in \{ 0, \ldots, n \}$...

2
votes

1
answer

101
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### 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{...

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91
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### 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 ...

3
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144
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### 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 $...

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52
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### 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} ...

4
votes

1
answer

112
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### 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 ...

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votes

2
answers

178
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### 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 ...

2
votes

1
answer

116
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### 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 (...