Questions tagged [norms]
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361 questions
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A question on the Banach space property of a rearrangement invariant function space
Consider a measure space $(S,\mu)$ and assume that $\mu(S)=1$. We consider the quantile function (or nonincreasing rearrangement) of a real valued function $f:S\to\mathbb{R}$ as the function
\begin{...
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On norms such that $(x_n)$ weakly convergent to $x$ and $\|x_n\|\to \|x\|$ imply $\|x_n-x\|\to 0$
Let us say that a norm $\|\cdot\|$ on $X$ is nice if each sequence $(x_n)$ weakly convergent to $x$ with $\|x_n\|\to \|x\|$ is norm convergent.
Using $\|x\|^2=\langle x,x\rangle$, it is easy to check ...
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When does the positive part of an entire function being in $L^2$ imply that the entire function is in $L^2$?
Let $f$ be an entire function of exponential type, real on the real axis, and $f^+$ its positive part, i.e.
$$
f^+(x) = \begin{cases}
f(x) & \text{if} f(x) \geq 0,\\
0 & \text{otherwise}.
\end{...
- 167
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What does the automorphism group of a norm cone look like in general?
Let $(V, \lVert \cdot \rVert)$ be a finite-dimensional real normed space, and let $C \subseteq \mathbb R \oplus V$ be the norm cone of $V$; that is, $C$ consists of all $(t, v)$ for which $\lvert t \...
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relation between the approximate $\gamma_2$ norms of Boolean matrix and sign matrix
For a real matrix $A$, its $\gamma_2$-norm is defined as
$$\gamma_2(A)=\min_{X,Y:A=XY}||X||_{row}||Y||_{col},$$
where $||\cdot||_{row},||\cdot||_{col}$ are defined as the maximum $\ell_2$-norm of row ...
- 551
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I am looking for "something like" an entry-wise matrix 1/2-norm. Has such a thing been studied? Where should I look?
Suppose an $n\times n$ square matrix $A$ with real or complex entries $A_{ij}$. Now define a quantity $Z(A)$ associated with the matrix by
$$
Z(A)=\sum_i \sum_j |A_{ij}|^{1/2}.
$$
What is this ...
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Has there been any study of normed semilattices?
First of all, I should confess that this is not my field of study. Recently, I encountered a situation where I had a (meet) semilattice $\mathcal{L}$ and a compatible valuation (positive semidefinite ...
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Is the maximal eigenvalue of this $n$ by $n$ matrix $\left\lfloor \log _2(n)\right\rfloor$?
The Mathematica 14 program for computing the matrix $T$ is
...
- 1,203
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Practical applications where one needs $L^p$ with $p\not\in\{1,2,\infty\}$
It appears to me that in practical applications one only ever needs the $L^1$, $L^2$ and $L^\infty$ norms, which are rather special cases among the $L^p$ norms. However, I am virtually sure that this
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- 3,768
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Equivalence between sum and integral of regular functions over positive real axis
Let $f$ be a function regular in $\Re z \geq 0$ whose indicator function satisfies:
$$h(\theta) = \limsup_{r \to \infty} \frac{\log |f(r e^{i \theta})|}{r} \leq c < \pi$$
in this half plane. Let $(\...
- 167
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Bound on norm of $H^2(D)$ functions outside the unit disk
Let $H^2(D)$ be the Hardy space of analytic functions on the unit disk $D$ with finite norm:
$$
\|f\|_{H^2(D)} = \sup_{r < 1} \left( \int_0^{2 \pi} |f(r e^{i \theta})|^2 \, d\theta \right)^{1/2}.
$$...
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Paley-Wiener-style theorem on Laplace transforms over Smirnov domains
My question is about the Laplace transform on the Smirnov space $E^2(\Omega)$ over a convex bounded Smirnov domain $\Omega$. The Smirnov space $E^2(\Omega)$ is the space of analytic functions on $\...
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On the extreme points of the unit ball associated to the Huber function
The Huber function is defined as follows
$$H(x) = \begin{cases} \frac{x^2}{2} & \text{if } x \leq \delta \\ \delta \cdot ( |x| - \frac{1}{2}\delta) & \text{otherwise} \end{cases}$$
The ...
- 2,602
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Is the unit sphere of a complex normed space similar to the unit sphere of a real normed space?
The James constant of a normed space $X$ is defined by
$$J(X)=\hbox{sup}\{\hbox{inf}\{\|x+y\|, \|x-y\|\}: \|x\|=\|y\|=1\}.$$
It is well-known that $J(X)=0$ or $J(X) \geq \sqrt{2}$.
On the other hand, ...
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Matrix norm inequality
I have two $n\times n$ matrices A and B where $A$ is positive-definite.
Is there any matrix norm such that if $||B|| \le c$, I could conclude $B^T A B \preceq c^2A$ in the Loewner order?
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