# Questions tagged [norms]

The norms tag has no usage guidance.

217
questions

**-1**

votes

**2**answers

124 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

183 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

141 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:\...

**1**

vote

**1**answer

163 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

108 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

272 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 ...

**2**

votes

**1**answer

66 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

**0**answers

132 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

188 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

46 views

### Reference request for a notion of $L^p$ norm of a one-form

I'm looking for a definition of $L^p$ norm for 1-forms on $\mathbb{R}^n$, with $n\geq1$. The first idea that I had was to set
$$
\left\Vert\omega_kdx^k\right\Vert_p:=\sum_{k=1}^n\left\Vert\omega_k\...

**0**

votes

**0**answers

74 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

52 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|\...

**6**

votes

**1**answer

405 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 ...

**1**

vote

**0**answers

48 views

### What is the distance of a particular root to the farthest one with respect to it as a function of a compacting factor?

Let $f:\mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function defined as
$$f := 2a_{1}(x - x^{p}) + 2a_{2}(x - x^{p})\sum_{k \in K}||x_{k} - x^{p}_{k}||^{2} + \dfrac{2a_{3}}{\alpha}\sum_{j \in J}\dfrac{...

**-1**

votes

**1**answer

73 views

### Holder inequality for a general rectangular matrix

Let $A \in \mathbb{R}^{m\times n}$ and $p,q \in \mathbb{R}^{+}$ such that $\frac{1}{p}+\frac{1}{q}=1$. I am interested to prove the following:
$$ \|A\|_{p}=\|A^T\|_q$$
I have tried using Holder ...

**0**

votes

**0**answers

61 views

### What is the closed-form solution to this double-sum norm function?

Given two points $A,B \in \mathbb{R}^2$, one defines the Euclidean distance $f: \mathbb{R}^2\times\mathbb{R}^2 \rightarrow \mathbb{R}^{\ge 0}$ as follows.
$$f(A,B) := \Vert A-B \Vert : = \sqrt{(A_{x}...

**3**

votes

**0**answers

108 views

### Lower bound on the intersection of $\ell_1$ $n$-balls

Let $B_1$ and $B_2$ be two balls in $\mathbb{R}^n$ in $\ell_1$ norm, with distance $d$ and radius $R$.
Is there a lower bound on the volume of the intersection between the two n-balls? (assuming the ...

**5**

votes

**1**answer

197 views

### What does the image of the integer lattice under a norm look like?

The question that I shall ask here has arisen in the context of Diophantine approximation. I find it rather interesting, and I have no idea how to answer it. Any help, advice, or suggestions for ...

**2**

votes

**0**answers

172 views

### Generalization of Ostrowski's Theorem

Let $\Lambda$ be a totally ordered set with two binary operations $+$ and $\cdot$ on it, such that:
1) $+$ is associative, has a unit $0$, and is symmetric.
2) $\cdot$ is associative, has a unit $1$,...

**2**

votes

**1**answer

184 views

### A problem in Bushnell and Henniart's book, “The local Langlands conjecture for GL(2)”

On page 123 of Chapter 5 in Bushnell and Henniart's book The Local Langlands Conjecture for GL(2), they state
an elementary property of tamely ramified extension of local fields, which is as follows,
...

**0**

votes

**0**answers

47 views

### Infer vector norm inequality from dominance in components

Let $p,q \in \mathbb{R}^n$ be non-negative, ordered vectors, i.e. $p_1 \geq p_2 \geq \dots \geq p_n \geq 0$ and similarly for $q$. Let
$$
||x||_r := \left(\sum_i^n |x_i|^r \right)^{1/r}
$$
for $(0,\...

**1**

vote

**1**answer

199 views

### How to calculate or estimate RKHS norm? [closed]

I am working with GP-UCB and need to calculate RKHS norm as in Theorem 6 of Srinivas et.al 2012. I found on page 3 column 1 like:
The induced RKHS norm $||{f}||_k=\sqrt{<f,f>}_k$ measures ...

**1**

vote

**1**answer

112 views

### Density of norm-attaining operators

By Bishop-Phelps theorem we know that for a real Banach space, the set of all norm attaining bounded linear functionals is norm-dense in $X^*$, the topological dual of $X$. We also know that in ...

**14**

votes

**1**answer

508 views

### What are the applications of the Mazur-Ulam Theorem?

Every bijective isometry between normed spaces is affine. This well-known and beautiful statement, the Mazur-Ulam Theorem, was proved in 1932, but the proof has been simplified and polished in years, ...

**3**

votes

**0**answers

96 views

### “Hoelder conjugate” version of the Johnson-Lindenstrauss transform

A variation of the well-known Johnson-Lindenstrauss transform (JLT) asserts that for $x_1,\ldots,x_m\in\mathbb{R}^n$ there exists a linear transformation $A:\mathbb{R}^n\to\mathbb{R}^k$ with $k=\...

**4**

votes

**1**answer

97 views

### Expected supremum of normalised random walk

Let $X^i\in \mathbb R^d$ be iid. random variables for $i=1$ to $n$.
Assume $\mathbb E[X^i]=0$ and the covariance matrix $\mathbb C[X^i] = \mathbb E[X^iX^{iT}] = I$ is the identity matrix.
Define $S^k=...

**7**

votes

**3**answers

694 views

### Norms as Points in $C(X)$

$\newcommand\abs[1]{\lvert{#1}\rvert}$Let $X$ be a compact hausdorff space, and put $C(X)$ for the $\mathbb{R}$-algebra of continuous maps from $X$ to $\mathbb{R}$.
For each point $x$, there is a ...

**0**

votes

**0**answers

59 views

### How to derive formula (10) norm to obtain formula (11) in Uncorrelated Group LASSO?

In Uncorrelated Group LASSO, Eq. (10) and Eq. (11) are as follows:
$J_2(W)=f(W)+\alpha Tr(W^TFW)$. (10)
$F_{ii}=\sum_{g}\frac{(I_{G_{g}})_i||W_{G_g}||_{2,1}}{||W_{G_g}^i||_2}$. (11)
where $w_{...

**1**

vote

**1**answer

68 views

### Matrix inequalities for the moment of the fixed Shatten norm

Let $A_i, i=1, \ldots, N$ be real (or complex) matrices of the same dimension. Let $r_i, i=1, \ldots, N$ be independent Rademacher random variables.
The following inequality gives a bound on the ...

**4**

votes

**0**answers

94 views

### $L^1$ norm of oscillatory integral operator

My question is about the $L^1_x$ norm of an oscillatory integral like
$$ \int_{\mathbb{R}^n} e^{i(y\cdot x+\lambda \phi(y))}f(y)dy,$$ where $\lambda \in \mathbb{R}$, $f\in C^{\infty}_c(\mathbb{R}^n)$ ...

**2**

votes

**0**answers

28 views

### Find the point that minimizes the summation of L_\infty norms to three given points

Given three points $\omega_1$, $\omega_2$, $\omega_3 \in \mathbb{R}^d$, how can I find the point $\omega \in \mathbb{R}^d$ such that the summation of its $\ell_\infty$ distances to these three points ...

**0**

votes

**1**answer

46 views

### Upper-bounds for a vector equation

Let $a$, $c$, $d$, $v \in \mathbb{R}^n$ are vectors, and $A, B \in \mathbb{R}^{n \times n}$ are matrices. Suppose that $ v = Ac-d $, and $a = ABc- \|B\| d$ where $\| B \|$ is the maximum value of the ...

**6**

votes

**1**answer

226 views

### Area of $n$-sphere contained outside $\ell_1$ ball

For a given $r>1$, what is the surface area of $\mathbb S^{n-1}$ (the sphere of radius 1 in $\mathbb R^n$) which is contained outside of the $\ell_1$ ball of radius $r$? Or equivalently, if $X\sim ...

**3**

votes

**1**answer

359 views

### Relation between Frobenius norm, infinity norm and sum of maxima

Let $A$ be a sequence of $n \times n$ matrices so that the Frobenius norm squared satisfies $\|A\|_F^2 \simeq n$ and the infinity norm squared is $\|A\|_{\infty}^2 = 1$. Is the following true?
$$\...

**5**

votes

**1**answer

2k views

### Relation between Frobenius, spectral norm and sum of maxima

Let $A$ be a $n \times n$ matrix so that the Frobenius norm squared $\|A\|_F^2$ is $\Theta(n)$, the spectral norm squared $\|A\|_2^2=1$. Is it true that $\sum_{i=1}^n\max_{1\leq j\leq n} |A_{ij}|^2$ ...

**3**

votes

**1**answer

144 views

### Gowers norms and three-term arithmetic progressions in the mean

Let $f:\mathbb{Z}^+\to \mathbb{C}$ be bounded. Say we are interested in studying how $f$ behaves in short three-term arithmetic progressions. It is very well-known that we can bound
$$\sum_{h\leq H} \...

**0**

votes

**1**answer

96 views

### Relationship between $2 \to 2$ norm and $\infty \to 2$ norm [closed]

I am wondering what are the best known relationship between $\|A\|_{2\rightarrow 2}$ and $\|A\|_{\infty\rightarrow 2}$ and how tight it is.
E.g., the trivial result is that for matrix $A$ with ...

**1**

vote

**1**answer

148 views

### Minimal value of matrix norm induced by a norm

Let $X$ be a finite dimensional Banach space and define a matrix norm $\| \cdot \|_{X}$ by
$$
\| A \|_{X} = \sup_{x \ne 0} \frac{\|A x\|_{X}}{\|x\|_{X}}
$$
where the matrix $A$ is interpreted as an ...

**4**

votes

**1**answer

334 views

### Bound for type of correlation measure

Assume you have a finite, discrete probability distribution for a joint random variable and such that $P(X=i,Y=j) = p_{i,j}$ for $i \in \{1, \dots, |X|\},j \in \{1, \dots, |Y|\}$. The marginal ...

**7**

votes

**1**answer

227 views

### Absolute value on tensor product of fields

Suppose that we have the Laurent series fields $F_1:=\mathbb F_p((X))$ and $F_2:=\mathbb F_p((Y))$.
Equip $F_1$ with the $X$-adic multiplicative absolute value $|\cdot|_1$, i.e. define $|X|_1=\dfrac{...

**4**

votes

**1**answer

265 views

### How sensitive are probability distributions to noise?

I'm trying to prove a result but I'm stuck at the very end of it: I'm having troubles understanding how noise propagates when considering a probability distribution. In other words, if I inject some ...

**1**

vote

**1**answer

155 views

### Characterizing a norm on sequences

Let $\{a_i\}$ be a sequence of reals such that $|a_i|\geq|a_{i+1}|$ for all $i$, and consider the following norm: $$\|\{a_i\}\| = \sup_k \frac{1}{\sqrt{k}}\sum_{i=1}^k |a_i|~.$$ One can see that -- ...

**7**

votes

**2**answers

314 views

### Proving an infinite norm minimization problem has finite support (non-convex p-norms)

Consider an optimization problem over infinite variables:
$$
\begin{align}
\min_{x}~& {\left\lVert{x}\right\rVert }_p
\\
\text{s.t}~& \left\langle x, a_n\right\rangle \ge 1~,~\forall n=1,\...

**0**

votes

**1**answer

357 views

### Total variation norm estimate

I have the following question concerning an estimate of the total variation norm. Let $\mu$ be a bounded Borel measure on $\mathbb{R}$ and denote by $\mu_t$ the measure defined by $\mu_t(\Omega):=\mu(\...

**0**

votes

**1**answer

227 views

### Does this norm have a specific name? Banach space? References?

Let $(X,\mathscr{B},\mu)$ be a $\sigma$-finite measure space. Let $\gamma$ be a probability measure on $L_2(\mu)$ with $\mathrm{supp} \, \gamma = L_2(\mu)$ and existing first moment. Then
$$
f \...

**6**

votes

**0**answers

242 views

### Find a condition such that the spectral radius for a special matrix is smaller than 1 (or a matrix norm smaller than 1)

We need a help to find a reasonable condition such that the spectral radius for a special matrix $\mathbf{J} \otimes\hat{\mathbf{G}}\hat{\mathbf{W}} + \mathbf{I}\otimes\mathbf{\hat{H}}$ is smaller ...

**3**

votes

**1**answer

213 views

### Largest eigenvalue of product of orthogonal-projection rank-1 perturbation

Suppose I have a symmetric positive definite matrix $A \in \mathbb{R}^{n \times n}$ with $n$ linearly indepedent columns $a_1,...a_n$ in $\mathbb{R}^n$. All columns $a_i$ has norm 1, but they are not ...

**0**

votes

**2**answers

279 views

### Existence of p=-infinity norm [closed]

Given a vector $\mathbf {x} =(x_{1},\ldots ,x_{n})$ the p-norm is defined as
$$
\left\|\mathbf {x} \right\|_{p}:={\bigg (}\sum _{i=1}^{n}\left|x_{i}\right|^{p}{\bigg )}^{1/p}
$$
for $p \geq 1$.
For $...

**3**

votes

**1**answer

91 views

### Sequence in *-algebra with different limits for two C*-norms?

The following question looks simple, but the answer is not obvious for me:
Let $S$ be a $*$-algebra and $\left\Vert \cdot \right\Vert _{1}$, $\left\Vert \cdot \right\Vert _{2}$ $C^*$-norms on $S$ ...

**1**

vote

**1**answer

146 views

### Is continuity preserved under norm operations

Let $F$ be a continuously differenable function over $\mathbb{R}$. Let $\Omega$ be
a bounded subset of $\mathbb{R}^2$. Assume that for every $w\in L^2(\Omega)$ then $v(x)=F(w(x))$, $x\in \Omega$, is ...