Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.
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106 views
Help me understand this proof (showing that something is a norm).
I am reading the following paper: Takáč, Peter On the Fredholm alternative for the p-Laplacian at the first eigenvalue. Indiana Univ. Math. J. 51 (2002), no. 1, 187–237.
I need help to understand the ...
5
votes
0answers
162 views
Monotone matrix norms
[Ciarlet 2.2-10]
Let $\mathscr{S}_n$ be the set of symmetric matrices and $\mathscr{S}_n^+$ the subset of non-negative definite symmetric matrices. A matrix norm $\|\cdot\|$ to be monotone if
...
4
votes
0answers
56 views
What are norms used for?
These two questions are quite similar to this one, so I apologise if this irritates anyone. Also, I suspect that a lot can be said in the answer, so I am really just looking for some main points ...
3
votes
0answers
150 views
Prove this equality in a Hilbert Space $H$ with Riesz's Representation Theorem.
For $x \in H$, prove that $\sup_{\|z\| = 1} \langle x , z \rangle= \| x \|$
Here is a quick one. If someone could improve this it would be great
Proof
By Cauchy Schwarz, $\langle x,z \rangle ...
3
votes
0answers
83 views
The norm of an operator
Let $\rho(x)$ be a weight function in a unit sphere, such that
\begin{equation}
\begin{array}{l}
\displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\
\displaystyle 2. \rho(x)\in ...
3
votes
0answers
52 views
Exercise from textbook about norm
The below diagram included a relevant page and the question 4 of the exercise which i am not sure how to do. Any hint or solution are welcome
Attempts: a) I have done it
b) i have tried to show that ...
3
votes
0answers
70 views
Absolute norms and 1-unconditional sums
Absolute norm
Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that
$$
\|(x,y)\|_N=N((\|x\|, ...
3
votes
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76 views
Is $\rho(f) := \mu(|f| \geq \epsilon), \epsilon >0$ some kind of generalized norm?
Let $\rho(f) := \mu(|f| \geq \epsilon)$ where $\epsilon >0$ is a constant and $\mu$ is a measure and $f$ is a measurable function. Then
$f\equiv 0 \rightarrow \rho(f) = 0$
when $|a| \neq 0$, ...
3
votes
0answers
149 views
Minimizing maximum absolute column sum norm of the residual between a matrix and its $k$-rank approximation
Let $X \in \mathbb{R}^{m\times n}$ be a matrix with rank $r$.
How can we find the optimal $\tilde{X} \in \mathbb{R}^{m\times n}$ whose rank is $k$ where $k\leq r$ and the reconstruction error in ...
2
votes
0answers
31 views
Proving norm inequality with Schwarz's inequality
I'm stuck on the following problem:
Suppose that $\{V_j : j \in \mathbb{Z}\}$ is a multiresolution analysis with scaling function $\phi$, and that $\phi$ is continuous and compactly supported. ...
2
votes
0answers
25 views
Are these two definitions for dual norm equivalent
Suppose there is a p norm, also is a function $\left\|x\right\|_p$, the dual function at $z$ of it is,
$$\sup\limits_x\left\{ z^{T} x-\left\|x\right\|_p\right\}$$
The second is,
$$
\sup\limits_x ...
2
votes
0answers
29 views
Finding an orthornormal basis given a bilinear form
Take $ \langle \cdot,\cdot\rangle$ to be a nondegenerate symmetric bilinear form (which may not be positive definite). Take $\Vert x\Vert = \vert \langle x,x\rangle \vert^{1/2}$. I want to show that ...
2
votes
0answers
39 views
Examples of functions that are Lipschitz w.r.t. Schatten p-norm?
A convex function $f$ is $R$-Lipschitz w.r.t. to a norm $\|\cdot\|$ if for all points $a, b$ we have $|f(a)-f(b)| \leq R\|a-b\|$.
For a real symmetric $n\times n$ matrix $A$ with eigenvalues denoted ...
2
votes
0answers
62 views
Is the polar set of convex Polytope also Polytope
Let $P$ be a convex polytope.
How can I prove that the polar set of $P$ (lets call it $P^*$) is polytope?
where $P^*=\{x\in\mathcal R^n:\forall v\in P, |\langle x,v\rangle|\le1\}$ .
$Thanks$
2
votes
0answers
148 views
Equivalence of norms is a equivalence relation
Two norms $||-||_1 $, $||-||_2$are equivalent if:
for two constants $a,b$ and $x$ from $V$ a vector space over a field it holds that : $$a||x||_1\leqslant ||x||_2\leqslant b||x||_1.$$
This is a ...
2
votes
0answers
72 views
Confusion about matrix norms
Reading the wiki article I get confused about matrix norms. My question, is it true that
$$\lVert Mx \rVert \leq C(\sup_{ij}{M_{ij}}) \lVert x \rVert$$
where $M$ is a matrix and $x$ is a vector and ...
2
votes
0answers
122 views
Bound on inverse operator
Define
$X = {C^{2, \alpha}}(U \times [0,T])$ and
$Y = {C^{0, \alpha}}(U \times [0,T])$ where $U$ is some real interval.
Let $F:X \to Y$ be a map. Let $DF(g):X \to Y$ be a bounded linear operator for ...
2
votes
0answers
98 views
How do you expand this frobenius form?
I'm trying to expand this Frobenius form $||C \circ (A-XB)||_F^2$ (here $\circ$ is the Hadamard point-wise multiplication). I want to find the minimum value with respect to X.
$$ ...
2
votes
0answers
100 views
Quick question: finite extensions and norms
[Edit: "Suppose K is a field complete with respect to a discrete valuation, with valuation ring $\mathcal{O}$."]
I'm trying to solve the following problem: let $L/K$ be a finite extension of fields, ...
1
vote
0answers
30 views
Find a matrix that minimizes the norm of a linear function
Let $S=AA^{T}+XX^{T}-AB^{T}\left(BB^{T}\right)^{-1}BA^{T}$ where $A$ and $B$ are known.
What is the best possible $X$
that minimizes $\left\Vert S^{-1}y\right\Vert _{2}$
for any vector $y\in ...
1
vote
0answers
44 views
why is $\ell_0$ a pseudo-norm?
Let $\mathbf{x}$ be a vector in $\mathbb{R}^n$. We define the $\ell_0$ pseudo-norm by:$$\|\mathbf{x}\|_0=\#\left\{i : \mathbf{x}_i\neq0\right\}$$
Why $\|\cdot\|_0$ is not properly a norm?
1
vote
0answers
54 views
Differences between a norm in $\mathbb{C}^n$ and a norm in $\mathbb{R}^n$
Consider the definitions of matrix norm and subordinate matrix norm from
Matrix Norm set #2
and let $A$ a real matrix and $\|\cdot\|$ a vector norm over $\mathbb{C}^n$. Define
\begin{eqnarray*}
...
1
vote
0answers
35 views
For any linearly independent vectors $x, y$, there is such a norm that $||x||_*>||y||_*$
Today I've seen in my class that:
For any linearly independent vectors $x, y$, there is such a norm that $||x||_*>||y||_*$
Our lecturer called it Benchmark theorem.
I wanted to learn more ...
1
vote
0answers
23 views
A modular which is not a metrizing modular (hence not an F-norm)?
I'm taking the terminology from Rolewicz's 1985 Metric Linear Spaces.
Given a complex vector space $X$, a modular $m$ is any function $m:X\to[0,+\infty]$ satisfying the following for all $x,y\in X$ ...
1
vote
0answers
76 views
Why is the negative entropy Lipschitz with respect to the $1$-norm (Over)?
Let $\left\|x \right\| = \sum_{i=1}^{i=n}\left|x^i\right|$ and $d\left(x\right)=\sum_{i=1}^{i=n}x^i\ln x^i$ where $x\in R^n $ and $ \sum_{i=1}^{i=n}x^i=1$
How to prove:
For all $x, x'$, $$\left| ...
1
vote
0answers
78 views
Hölder norm estimates
How do you prove the following estimate for composition of functions:
If $k\geq 1$, then there exists a constant $c=c(k,\alpha)$ such that
$$
\|f_1\circ g_1-f_2\circ g_2\|_{k,\alpha}\leq ...
1
vote
0answers
43 views
Optimizing over norms of set of equations.
I have a set of N real-valued finite-dimensional vectors $\mathbf{v}_i$ and target norms $y_i$ and I am trying to find a linear transformation matrix $L$ such that the norm of the transformed vectors ...
1
vote
0answers
117 views
Step functions dense in Integrable functions with respect to $L_2$
Let $I$ be a bounded interval. Prove that $\{\text{step functions }I \to C\}$ is dense in $\{\text{integrable functions }I \to C\}$ (Riemann Integrable) with respect to $\|.\|_2$ ($L_2$ norm)
1
vote
0answers
88 views
Why is the subdifferential of norm of a matrix ||A|| defined like this?
I read in a paper called "Characterization of the subdifferential of some matrix norms"
that it defines the subdifferential of the matrix norm like this:
$$\partial ||A||=\{G \in R^{m\times n} : ...
1
vote
0answers
61 views
How to represent and handle eigenvalue constraint in optimization
For example consider the problem
$\min_X f(X)$
s.t. $\lambda_i(X+A)=\lambda_i(B)$ for $i \in {1,...,N}$
where $A$ and $B$ are full rank N by N matrix, $\lambda_i(X)$ is the i-th eigenvalue of $X$
...
1
vote
0answers
100 views
Upper bound for L1-L2 optimization problem
I am interested in the following convex optimization problem:
\begin{align*}
\max & ||x||_1 \\
\text{s.t.} & ||x-a||_1 \le K \\
& ||b\circ x||_2 \le 1\\
& x \in R^n
\end{align*}
where ...
1
vote
0answers
116 views
Bound on euclidean norm
Is it possible to find a suitable lower bound on
$$\left|\left(tM+\sum_{k=2}^\infty\frac{(tM)^k}{k!}\right)\cdot b\right|$$
for $M$ as $n \times n$ matrix, $b$ as $1 \times n$ vector and for all $t$ ...
1
vote
0answers
84 views
How to prove an inequality for a special structure of strictly triangular matrix
The problem I cause is attached below. I am trying to prove the inequality.
By using small $M$, I found that the terms on the left side of the inequality are part of the terms of the expansion on ...
1
vote
0answers
131 views
Dual Norm Of Sum Of Norms
What is the dual of a norm that is the sum of two-norms? Specifically, say we have the following norm for a $\mathbf{x}\in \mathbb{R}^n$ and $\mathbf{A}_i \in \mathbb{R}^{m \times n}$
$\|\mathbf{x}\| ...
1
vote
0answers
165 views
The geometry of functions $\mathbb{R}^2\rightarrow \mathbb{R}$ that satisfy the norm axioms
What are constraints on the "looks" of a function $f:\mathbb{R}^2\rightarrow \mathbb{R}$, if $f$ satisfies $$f(x_1+x_2)\leq f(x_1) + f(x_2), \ \quad x_1,x_2\in \mathbb{R}^2 \quad \quad (1)$$i.e. the ...
1
vote
0answers
75 views
Inequality estimation
Let $B$ be an open unit ball in $\mathbb{R}^d$ centered at the origin and $u$ be a twice continuously differentiable function on $\bar{B}$ with $u|_{\partial B} = 0$.
Know
$$\Delta u = f.$$
How can I ...
1
vote
0answers
78 views
constrained optimization of dot product
Given a real matrix $A$ find a positive vector $x$ of unit length ($x^T x = 1$) for which $x^T A^T A x$ is minimal (closest to $0$).
A has size about $1000 \times 20$ and can be written as $[ A_P | ...
1
vote
0answers
98 views
Help with a proof that the inversion mapping for matrices is continuous
I'm reading a proof that the map $A\mapsto A^{-1}$ is continuous in the operator norm. A part of the proof is that if $A,B$ are matrices such that $A$ is invertible and $\beta=\parallel ...
0
votes
0answers
32 views
Formulas involving the square of a norm
Why does $$\|x-y+\alpha z\|^2=\|x-y\|^2+2\alpha\langle x-y,z\rangle+\alpha^2 \|z\|^2$$
but
$$\|x-z+\theta z-\theta y\|^2=\|x-z\|^2+2\theta\langle x-z,z-y\rangle +\theta^2 \|z-y\|^2?$$
Why is there ...
0
votes
0answers
15 views
Confusion related to spectral norm
I was reading this paper where they gave the error results in spectral norm form. If C is the actual matrix and $\hat{C}$ is its estimate. The spectral norm of $||C-\hat{C}||_{l2}$ what does it ...
0
votes
0answers
39 views
Calculation of the sub gradient of the first norm of a matrix
Lets say I have a matrix X and its first norm $||X||_1$. How do I calculate the subgradient of this norm with respect to matrix X itself.
0
votes
0answers
20 views
Condition number of a function
I would like to find the Condition number of a function (f(x)) with one variable (x) and several parameters. which can be calculated by:
$$ c(x) = ||x||*||f'(x)||/||f(x)||$$
Here if my function is ...
0
votes
0answers
26 views
L1 penalty can serve as a convex surrogate for an L0 penalty. Why?
I have heard machine learning practitioners say that the $L_1$ penalty is a (or can serve as) convex surrogate for an $L_0$ penalty (in the context of optimization and statistical fitting).
What do ...
0
votes
0answers
40 views
Is it general to say “norm” to mean 2-norm when it is on an inner product space?
Let $V$ be an inner product space over $\mathbb{F}$.
If one defines $\lVert \bullet \rVert$ as $\sqrt{\langle \bullet, \bullet \rangle}$, then $\lVert \bullet \rVert$ is a norm on $V$.
However, if ...
0
votes
0answers
36 views
Subgradient of matrix $l1$-norm
Let $X$ be a square matrix, what is the sub-gradient of $f(X) = ||AX-XB ||_{l1}$?
$A$ and $B$ are both constant matrix.
I am very confuse about the chain rule on matrix derivatives.
0
votes
0answers
39 views
Matrix norm applications?
There'are many different ways to calculate Matrix norm.
But once calculated, what is the practical use/application of it (e.g. in computer programming)?
Or does it let define something that can be ...
0
votes
0answers
35 views
All matrix/vector norms induce the same topology?
From Wikipedia
all norms on $K^{m \times n}$ are equivalent; they induce the same topology on $K^{m \times n}$. This is true because the vector space $K^{m \times n}$ has the finite dimension $m ...
0
votes
0answers
20 views
Relation of matrix / operator norm to rank deficiency?
(This is a follow-up to Criterion for detecting rank-deficiency via QR decomposition?) To recap:
I'm solving a system like $P \approx X Y^T$, where P is a large sparse
matrix, and X and Y are a ...
0
votes
0answers
27 views
Subgradient and Lipschtz
For a convex function $f:R^n\longrightarrow R$,
the function is G-Lipschitz with any norm x
$\left| f\left(w\right)- f\left(w^{'}\right)\right| \leq G \left\|w-w^{'}\right\|_x$ ,
if and only if
...
0
votes
0answers
42 views
Boundary value problem domain, norm?
I'm starting some research on a boundary value problem for a memoire for the 2nd semester of a masters, and my mentoring professor asked me to find the answer to the following question:
For $u \in ...
