Questions tagged [norms]
The norms tag has no usage guidance.
279
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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
$$ \...
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2
votes
1
answer
89
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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 ...
- 23
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31
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Proving sup-norm of a specific polynomial (Kaisjer--Varopoulos, 1974)
This question regards a proof in the addendum (due to Kaisjer and Varopoulos) to "On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory,&...
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Are more duality mappings for matrix norms known?
When reading A Unifying Representer Theorem for Inverse Problems and Machine Learning by Michael Unser and Duality Mapping for Schatten Matrix Norms by his PhD student
Shayan Aziznejad, I wondered if ...
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2
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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 : ...
- 489
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76
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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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1
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67
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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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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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0
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89
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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 $i \in \{ 0, \ldots, n \}$...
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1
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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:
$$
\begin{aligned}
\text{mode of x} = \arg \min_s \sum_i \lvert x_i - s \rvert^0 \\
\text{...
- 123
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votes
0
answers
60
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A truncated Frobenius norm of a matrix is convex or not?
Given a positive integer $k$ and a matrix $X\in \mathbb{R}^{m\times n}$. A truncated frobenius norm of a matrix $X$ is defined by
$$\Vert X \Vert_{k,F} = \sqrt{\sum_{i=k+1}^{m} \sigma_i^2(X)},$$
where ...
- 143
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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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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} ...
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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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2
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140
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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 ...
- 431
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1
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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 (...
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Strong convexity inequality w.r.t. infinity norm $\lVert\cdot\rVert_{\infty}$
Consider a twice differentiable 1-strongly convex function $f:\mathbb{R}^n \to \mathbb{R}$.
Is it true that there exists $\alpha>0$ independent of $n$ such that, for all $x \in \mathbb{R}^n$:
\...
- 33
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0
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bound norm of algebraic integers in cyclotomic field
Let $\zeta$ be the $p$th root of unity, with $p$ an odd prime number.
Let $\mathbb{Q}(\zeta)$ be the $p$th cyclotomic field and let $\mathcal{O}=\mathbb{Z}(\zeta)$ the ring of integers of $\mathbb{Q}(\...
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Is a completion of strictly convex normed space strictly convex?
A (real) normed space $(V, \lVert \cdot \rVert_V)$ is called strictly convex if for all $x, y \in V \setminus \{ 0 \}$ we have
\begin{equation}
\lVert x + y \rVert_V = \lVert x \rVert_V + \lVert y \...
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2
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If I multiply the coefficients of a trace-class operator with bounded complex numbers is it still trace class?
Suppose that $T \in TC(l^2( \mathbb{Z}))$ is trace class.
Consider its kernel $ T(i,j) = \langle e_i, T e_j \rangle $ where $ \{e_i\}_{i \in \mathbb{Z}}$ is an ONB for $l^2( \mathbb{Z})$. Now, ...
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A pullback in the norm of the Sobolev space $H^{-\frac 1 2 }(\Gamma)$
Let $A$ be a $3\times 3$ real constant symmetric positive definite matrix, $\Omega\subset\mathbb{R}^3$ a bounded Lipchitz domain with boundary $\Gamma$, $\Omega'=A^{-\frac 1 2}(\Omega)$ (so we have $\...
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Characterization of differentiability
For a normed space $(V, \lVert\cdot\rVert_V)$ let us define:
\begin{equation}
\forall x, y \in V \quad [0,1] \mapsto \gamma_x^y (t) = (1-t)x + ty.
\end{equation}
I would like to ask whether the ...
- 431
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vote
2
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273
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Lp norm of Hadamard matrix
What is the Lp norm of the $N$-dimensional Hadamard matrix $H = ((-1)^{i \cdot j})_{i,j}$ for $p > 2$? I know that $\|H\|_1 = N$, $\|H\|_2 = \sqrt{N}$, $\|H\|_\infty = N$ but I can't figure out ...
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When does a finite metric induce a matrix norm?
If I have a metric $d(\cdot,\cdot)$ on the set $\{1,\dots,n\}$, are there well-known necessary or sufficient conditions for the existence of a matrix norm $Q$ that induces that metric on the unit ...
- 3,668
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0
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Can one extend the norm function to the symmetric square of a (complexified) Clifford algebra?
Let $A = Cl_{r,s} \otimes \mathbb{C}$ be the complexification of the real Clifford algebra $Cl_{r,s}$ associated to a non-degenerate quadratic form on $\mathbb{R}^n$, with $n = r+s$, with signature $(...
- 3,594
1
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1
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Norm contrained Gaussian distribution
Let $X$ be a multivariate normal $\mathcal{N}(\mu, \Sigma^2)$ and let $X$ be anisotropic, that is I am considering $\Sigma$ to be a diagonal matrix but the elements on the diagonal might be different.
...
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Product absolute value in rings of integers
Let $F$ be an algebraically closed field of characteristic $p$ equipped with a nonarchimedean dense absolute value $|\cdot|:F \rightarrow \mathbb{R}_{\ge 0}$ with respect to which $F$ is complete. Let ...
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Continuously varying norms
Let $V$ be a finite-dimensional real vector space with its Euclidean topology. Then all norms on $V$ are equivalent and consequently given two norms $\lVert-\rVert$, $\lVert-\rVert'$, the number
$$
d =...
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What are the functions such that $ \lVert f + g \rVert_p^p = \lVert f \rVert_p^p + \lVert g \rVert_p^p$?
Let $1 \leq p \leq 2$. I am looking for a characterization of the couples $(f,g)$ of functions $f,g \in L_p(\mathbb{R})$ such that
$$ \lVert f + g \rVert_p^p = \lVert f \rVert_p^p + \lVert g \rVert_p^...
- 2,010
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1
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Estimate of Hölder Norms (Littlewood–Paley theory)
I'm currently studying Littlewood–Paley theory and its application to norm estimate/PDEs by reading Muscalu and Schlag's textbook, where I encountered the following norm estimate problem:
Recall that ...
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Polyhedron coordinate bound
Given a polyhedron
$$Ax\leq b$$
where we assume $A\in\mathbb Q^{m\times n}$ and $b\in\mathbb Q^{m}$ and it takes $L$ bits to represent the inequalities what is a good bound on the quantity $\|y\|_\...
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Green's identity with a different norm
Let $\Omega \subset \mathbb{R}^n$ be a domain with a smooth boundary $\Gamma$. Suppose that $f, g \colon \mathbb{R}^n \to \mathbb{R}$ are of class $C^\infty( \overline{\Omega})$. Then Green's first ...
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$L^p$ estimate of a multiplier operator
I'm studying harmonic analysis by myself and I encountered the following claim about multipliers: consider a sequence of complex numbers $\{m_{n}\}_{n \in \mathbb{Z}}$ that satisfies:
$$\sum_{n \in \...
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3
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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 ...
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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 ...
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1
answer
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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 $C=C(s)>1,$...
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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 $\|\...
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452
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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 ...
- 13.2k
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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,
$...
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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
$$(\sigma_1=|\lambda_1|)\geq (\sigma_2=|\lambda_2|)\geq \dots \geq (\sigma_r=|\...
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Condition number for matrix of eigenvectors of a diagonalizable matrix
Let $A$ be a diagonalizable matrix, i.e., $A=SDS^{−1}$. For any matrix $A$, condition number is defined as $\kappa(A)=\|A\| \|A\|^{-1}$.
For $A$ being a diagonalizable matrix, define $G_A=\{{S: S^{-1} ...
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Decomposition of the Orlicz norm into sequential norm
I am bearing seeking for a sequential decomposition of the norm in Orlicz space.
Let me state what is known in the particular case of Lebesgue space $L^p(\Bbb R^d)$.
Given $u\in L^p(\Bbb R^d)$ let
$$n\...
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1
answer
64
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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:
$x,y \in \mathbb{R}^n$,...
- 19
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votes
1
answer
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Uniform smoothness inequality for Schatten norms
I've previously asked this question on stack exchange.
I'm looking for a proof of the inequality
$$
\left[ \frac12(\left\|A+B\right\|_p^p + \left\|A-B\right\|_p^p)\right]^{2/p} \leq \left\|A\right\|_p^...
- 83
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Estimate of the norm of the radial part of a function
Consider a function $u\in L^2(\mathbb R^N)$, and another function $\varphi$ which is the unique solution to the Poisson equation $\Delta \varphi = u$ vanishing at $\infty.$ We know that the radial ...
- 1,457
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1
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Norm of a matrix with clustered eigenvalues
On page 271 of Trefethen and Bau's Numerical Linear Algebra, it is constructed a matrix
$$A=2I_{m\times m}+0.5\cdot\frac{\text{rand}(m)}{\sqrt{m}}$$
for $m=200$, where rand(m) is an array with $m\...
- 13
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3
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Bounding supremum norm of Lipschitz function by L1 norm
Consider $f:[0,1]^d \to \mathbb{R}$. Suppose that $f$ is $L$-Lipschitz w.r.t. the Euclidean norm. Can we provide an upper bound on $\|f\|_\infty$ in terms of $\|f\|_1 := \int_{[0,1]^d} |f(x)|dx$ ?
In ...
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Subtracting the weak limit reduces the norm in the limit
Question
Let $X$ be some reflexive Banach space. Suppose $x_n$ is some sequence in $X$ that weak converges to some $y \neq 0$. Is it the case that
$$ \limsup \|x_n - y\| < \limsup \|x_n\| ?$$
...
- 30.7k
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2
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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\...
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