Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.

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8
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2answers
166 views

$L^2$ Bounds for Markov Chains.

Consider a non-negative, square stochastic $n \times n$ matrix $P$ (rows sum to one, $P$ is ergodic). We are interested in characterizing the set of $n \times n$ invertible matrices $A$ such that we ...
2
votes
1answer
56 views

Norm of a linear functional, sup, definite integral

I'm trying to find the norm of this mapping: $$\phi: C^1([0,1]) \ni f \rightarrow \int_0^{1/2} f(t)dt + f'(\frac{2}{3}) \in \mathbb{R}$$ with $||f|| = \sup_{t \in [0,1]}|f(t)| + \sup_{t \in ...
6
votes
1answer
36 views

Integral with absolute value of the derivative

I'm trying to estimate this integral $\int_0^1 t |p'(t)|dt$ using this value $\int_0^1 |p(t)|dt$; here $p $ is a real polynomial. This means, I am looking for an $M>0$ such that $$\int_0^1 |t ...
1
vote
1answer
51 views

Polynomial, bounded functional

In order to prove continuity of the functional $$\varphi: \mathbb{R}[X] \ni p \rightarrow p'(2011) \in \mathbb{R}$$ where $$||p|| = \sup \{|p(t), \ t\in [0,1]\}$$ I'd like to prove that this ...
0
votes
0answers
180 views

L0 norm, L1 norm and L2 norm

For vector $\boldsymbol{x},\boldsymbol{y} \in \mathcal{R}^{n}$, if \begin{equation} \| \boldsymbol{x} \|_0 = \| \boldsymbol{y} \|_0 \end{equation} What relationship will $\| \boldsymbol{x} \|_1$ and ...
0
votes
1answer
40 views

Why is the euclidean norm not differetiable at $0$?

I denote $N(x)$ as the norm-function, although in the denominator it stays $\|x\|$. $$\lim_{x\to 0} \frac{N(x)-N(0)}{\|x\|} = \lim_{x\to 0} \frac{N(x) - 0}{\|x\|} = \lim_{x\to 0} 1 = 1 \ne 0$$ 1) ...
3
votes
2answers
296 views

Proof that the Euclidean norm is indeed a norm

I apologize beforehand for this question. Its embarrassing I know. Anyway, here we go. Recall: $$ \| x \|^2_{\mathbb{R}^2} = \sum^{n}_{i = 1} x^2_{i}$$ How do we prove its a norm? Well if its a ...
0
votes
1answer
47 views

lower bound on matrix norm inequality of sum

The question is simple: can we say this? $\|A\|-\|B\|<\|A+B\|$ for any norm you like.
1
vote
2answers
35 views

Norm of Functional on : $c_0 = \{ x \in l^\infty :\ \lim_{k \to \infty}=0 \}$

Take $E = c_0 = \{ x \in l^\infty :\ \lim_{k \to \infty}=0 \}$ and define on $E$ the functional: $$F(x) = \sum_{n=1}^\infty \frac{x_n}{2^n}$$ $\cdot$Show that $F$ is a linear continuos functional on ...
0
votes
1answer
44 views

Norm of Integral Operator on $E=\{u \in C[0,1]\ ,\ u(0) = 0\}$

There are similar question but the characterization of the space $E$ that I have gives me problem in computing the actual norm. Let $E=\{u \in C[0,1]\ ,\ u(0) = 0\}$ with the usual $\parallel ...
0
votes
0answers
62 views

when does the equality hold for the matrix norm triangle inequality and product inequality

So here is the problem: When does the equality hold for the following two famous "matrix" norm inequalities: $\|A+B\|\leq \|A\|+\|B\|$ $\|CD\|\leq\|C\|\|D\|$ For any norm you prefer. But I'm ...
0
votes
0answers
34 views

solver for non-convex matrix optimization with convex constraints

So here is the problem: $\max_{D} ~~ \|A+BD\|$ subject to $\|D\|<1$ (any norm you like) where matrices A and B are given. The cost function is evidently convex as well as the constraint, but ...
2
votes
1answer
38 views

$A\in \mathbb{R}^{n\times m}$, proove that the sum of the maximum of each row is less than $n\|A\|_1$

I'm quite sure this is true, but I'm having trouble in finding a proof for $\mathbb{R}^{n\times m}$. I already found it for $\mathbb{R}^{2\times 2}$. Let $A\in \mathbb{R}^{n\times m}$, such that: ...
2
votes
3answers
34 views

Hölder inequality in case $q=p=2$.

It should return the Cauchy-Schwarz inequality, but I'm having trouble with comparing the left sides of the inequalities: For example, if $x=(4,3)$ and $y=(3,-4)$, then $\sum_{v=1}^2 |x_vy_v| = ...
2
votes
2answers
55 views

Interpreting norm definition

Book: Convex Optimization (Author: Stephen Boyd), Appendix A, Topic: A.1.2 Norm,distance, and unit ball Can anyone please help me in understanding the following definition of "norm" $$ \| x \| ...
1
vote
1answer
37 views

Is it true that $\|M(I-M)^{-1}\|_{\infty} \leq \frac{\|M\|_{\infty}}{1-\|M\|_{\infty}}\,?$

I've encountered a non-symmetric matrix $M$ with nonnegative elements ($M_{ij} \geq 0$), satisfying $\|M\|_{\infty} < 1$, which I need to bound. Is it true that $$ \|M(I-M)^{-1}\|_{\infty} \leq ...
1
vote
2answers
61 views

A condition equivalent to orthogonality

Prove that in any inner product space: $x$ and $y$ are orthogonal if and only if $||x+\alpha y||\ge ||x||$ for every scalar $\alpha$.
1
vote
1answer
62 views

Spectral norm upper bound for covariance matrix

Let $\|\cdot\|_2$ be the spectral norm. Let $x_1,\dots,x_n$ be i.i.d. draws from $N(0,S)$. Let $\lambda_1,\dots,\lambda_n$ be some real numbers. Is it true that $$\|\sum_{i=1}^n \lambda_i x_i ...
3
votes
1answer
94 views

Relation between Frobenius norm and eigenvalues

I'm considering a stochastic multivariate process, the stability of which implies that all eigenvalues $\lambda_i$, $i = \overline{1,n}$ of a certain square real-valued matrix $A$ lie within the ...
0
votes
1answer
31 views

Bound a Lyapunov storage function

How to effectively bound the following entity to deduce its definite negativeness $\dot{v} = -k_1 e_1^\top A e_1 + k_1 e_1^{\top} A e_2- k_2|e_2|^2$, with A a positive definite square matrix, $e_1$ ...
0
votes
1answer
42 views

Matrix norm properties: inequalities

Is the fololwing relationship always true: $x^{\top} (A - \frac{1}{2} \|A\|_F \, I_3) x > 0$, knowing that matrix A is definite positive?
1
vote
1answer
74 views

Question about a proof in Rudin's book - annihilators

I'm reading the proof of the following theorem in Rudin's "Functional Analysis": Let $M$ be a closed subspace of a Banach space $X$. The Hahn-Banach theorem extends each $m^* \in M^*$ to a ...
1
vote
0answers
60 views

Max - min problem of a quotient of norms

For the $2\times2$ matrix $\begin{bmatrix}4&0\\-3&-5\end{bmatrix}$ Part 1 Find nonzero vectors $u$ and $w$ that maximize and minimize respectively the quotient $||Av|| / ||v||$. Part 2 ...
2
votes
1answer
58 views

Which is finer(larger) between the sequence spaces $l_{p}$ & $l_{p+1}$

Prove that, $l_{3}\subset l_{7}$ & $L_{9}[0,1]\subset L_{6}[0,1]$, where $l_{p}$ & $L_{p}[0,1]$ are of usual notation. Are the converses hold for both cases? Can these two results ...
0
votes
1answer
65 views

Verifying that the Sobolev space is a Banach Space

In PDE Evans 2nd edition (pages 262-263), I am trying to understand a proof of a theorem, which states: THEOREM 2 (Sobolev spaces as function spaces). For each $k=1,\ldots$ and $1\le p\le\infty$, ...
0
votes
1answer
29 views

matrix F-norm inequalities on matrix sum and product

As you know, we have the two following inequalities for sum and product (for Frobenius norm): $\|A+B\|_F\leq\|A\|_F+\|B\|_F$ and $\|AB\|_F\leq\|A\|_F\|B\|_F$. The question is, under which ...
2
votes
0answers
47 views

Reference for an inequality between $|x+y|^p$, $|x|^p$, $|y|^p$, and $|x-y|^p$

I am interested in an inequality for real numbers $x,y$ and $1<p<\infty$ that it should say something like $$ |x+y|^p -|x|^p - |y|^p \leq (1-2^{1-p})|x-y|^p. $$ Is this inequality (or ...
1
vote
0answers
48 views

How nuclear norm is convex whereas weighted nuclaer norm is not?

In (http://nuit-blanche.blogspot.in/2014/05/wnnm-weighted-nuclear-norm-minimization.html), it is stated that nuclear norm of a matrix $\mathbf{X}$, given as $||\mathbf{X}||_{*}=\sum_{i} ...
0
votes
1answer
58 views

Cauchy Schwarz inequality and absolute value

Here's the inequality: $$|\langle u,v\rangle|\le\|u\|\cdot\|v\| $$ Why on the LHS there's an absolute value? We know that $\langle u,v\rangle \ge 0$ Isn't it redundant?
0
votes
0answers
50 views

Product spaces $X = Y = \mathbb R$

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. Let $d_{X \times Y} : X \times Y \rightarrow \mathbb R_+$ be given by $$d_{X \times Y}((x_1,y_1),(x_2,y_2)) := d_X(x_1, x_2) + d_Y(y_1, y_2)$$ How ...
0
votes
1answer
32 views

Dirichlet problem, solvability

Let $\Omega\subset \mathbb R^n$ and $f\in L^2(\Omega)$. We consider $\{u\in C^1(G):||u||_{1,2,G}<\infty \}$,where $||u||_{1,2,G}$ is the "sobolev-norm" with parameter $1,2$ on $G$. Then $H^1(G)$ ...
2
votes
1answer
51 views

Inequality$\Big|\sum_{j=1}^n a_{1j} x_j \Big|^2 \leq \sum_{j=1}^n |a_{1j}|^2 \sum_{j=1}^n |x_j|^2$

Let ${\bf A}$ be a $m \times n$ matrix with entries $a_{ij}$, and ${\bf x}$ be a $n \times 1$ vector with entries $x_{i}$. Then how can I show $$ \left\vert\,\sum_{j\ =\ 1}^{n} ...
2
votes
0answers
52 views

Norm for a set of vectors

Let V be a normed vector space (real or complex valued) with norm $\|\cdot\|_V$. For any nonempty and bounded subset $A \subseteq V$ one can define $\|A\|$ via $$\|A\|:=\sup\{|x|:x\in A\}$$ I ...
0
votes
1answer
20 views

Product of $L_2$ norm of vectors

Is the $\sum \Vert b_k\Vert_2^2 \le\ge= \sum \Vert b_k\Vert_2^2 \Vert a_k\Vert_2^2$ ? where $b_k$ is a column vector and $a_k$ is a highly sparse row vector.
0
votes
1answer
80 views

Frobenius Norm to L2 norm conversion

Can anyone help me with a relation: I want to represent Frobenius norm of matrix $X$ in terms of l2 norm of constituent vectors $b$ and $a$ $\Vert X\Vert_F^2=\Vert ba \Vert_F^2$ where $b$ is a ...
0
votes
0answers
15 views

norm bounded Convolution in matrix space

There is a stable matrix $A$ with eigen values in unit circle,for discrete time system : $x(k+1)=Ax(k)+f(k)$ can we prove: $||\Sigma_{j=0,..,k} A^{k-j}f(j)||_2<= ||f(k)||_2/{(1-A_{max})} $ where ...
0
votes
1answer
35 views

Trace distance between “weighted” Hermitian matrices

The trace norm for a matrix $\mathbf{A}$ (also known as Shatten 1-norm) is defined as follows: $\|\mathbf{A}\|_1=\operatorname{trace}[\sqrt{\mathbf{A}^*\mathbf{A}}]$. It yields a useful distance ...
3
votes
1answer
40 views

Norm of an operator and eigenvalues

I have $K\colon L^2(0,T) \to L^2(0,T)$ a Hilbert-Schmidt integral operator (and so $K$ is linear, bounded, compact and self-adjoint) and I have obtained its eigenvalues and eigenvectors. From them, I ...
1
vote
2answers
50 views

Polynomials of best approximation

The question is about approximating the continuous function in an interval $[a, b]$. If we consider the linear space of all such functions endowed with the norm $$||f|| = \max_{x \in [a, b]}|f(x)|$$ ...
1
vote
1answer
32 views

Unit ball for a special norm

What does the unit sphere for the norm on $\mathbb{R}^2$, $\displaystyle N(x,y)\rightarrow\sup_{t\in\mathbb{R}}\frac{|x+ty|}{t^2+t+1}$, look like ? My approach was to consider $y=ax$ so as to get ...
4
votes
2answers
111 views

Minimizing the Frobenius Norm

I would like to minimize the following expression with respect to matrix X: $$\left \| A-BX \right \|_{F}$$ where A and B matrices are given and all the matrices have positive integer elements. Any ...
0
votes
0answers
26 views

Norm of linear functional in $W^{1,2}$

I want to find the norm of the following linear functional: $\phi:W^{1,2}_{[0,1]} \rightarrow\mathbb{R}$ $\phi(f)=\int^1_0t\cdot f(t)+ f^{'}(t) dt $ The norm is: ...
0
votes
2answers
86 views

Maximum eigenvalue of product of two matrices

Let $A$ and $B$ be two Hermitian matrices. I wanted to know if there is any relation between the maximum eigenvalue of $AB$ and that of $A$ and $B$. Is the following relation true? ...
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0answers
41 views

Greatest norm for the exponential of a polytopic matrix

im new to the forum and I have a question that Im working quite for a while now. I would like to prove that the greatest norm when considering the exponential ...
1
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0answers
80 views

Matrix norm induced by a vector norm.

All matrices are real. $A$ is a matrix of size $n \times k$ with $k < n$ and has independent columns. The function $v(x) = \|Ax\|_1$ is a norm. What is the matrix norm induced by $v$? Is it of ...
0
votes
1answer
52 views

What does it mean to write $|||x|||$ rather than $||x||$?

I am familiar with the notation $||x||$ meaning some norm of $x$. I have just come across the notation $|||x|||$ (in a text that also uses the former for norms). What is the difference?
2
votes
2answers
120 views

$\| AB\|_\square \leq 1$ implies $\| BA\|_\triangle \leq 1$

In this post norm denotes a matrix norm, i.e. it is sub-multiplicative. All matrices are real. $A$ is of size $n \times k$ with independent columns ($k \leq n$). $B$ is of size $k \times n$. Let $\| ...
1
vote
1answer
93 views

What is the general definition of the conjugate of a multiple-component number?

I know that the conjugate of a binomial is the negation of the second part. So the conjugate of (a + b) would be (a - b). I know that the conjugate of a complex number (a + bi), similarly, is (a - ...
5
votes
1answer
64 views

A question concerning the Schwartz space

Denote the Schwartz space by $\mathcal S(\mathbb{R})$. I want to show that $\forall n,k \in \mathbb{N} \cup \{0\}$, $\|\cdot\|^{(n,k)} : \mathcal S(\mathbb{R}) \rightarrow [0, \infty)$ defined by ...
1
vote
1answer
51 views

$\|u\| = \Big(\int |\nabla u|^p \Big)^\frac{1}{p}$ as a norm on $W_0^{1,p}(\Omega)$ where $\Omega$ is bounded

Let $\Omega$ be a bounded open set of $\mathbb{R}^N$, define $\|u\|_1 = \Big(\int |\nabla u|^p \Big)^\frac{1}{p}$ as a norm on $W_0^{1,p}(\Omega)$ for $2\leq p<\infty$, where $$\int |\nabla u|^p ...