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

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Unique minimizer of $\|x\|_{\mathcal A}$ subject to $\Phi x=\Phi x_0$

I'm trying to understand the proof of Lemma 2.3 of the paper Simple bounds for recovering low-complexity models. The authors want to find bounds on the numbers of rows $m$ of $\Phi$ to ensure that ...
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1answer
8 views

Gradient of the function $f(x)= \|x\|^p$

How can the gradient in this case be computed? I understand that $f(x)= (x_1^2 + x_2^2 + \ldots + x_N^2)^{p/2}$ but how do I proceed from here?
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135 views

For what $X$ do we have $\|XA\|_{1} \leq 1$ for a given $\|A\|_{1} \leq 1$.

All matrices are real. By $\| \cdot \|_1$ I mean a matrix norm induced by the vector norm $L_1$, i.e. the max of the column sums of absolute values. The matrix $A$ is given and we have $\|A\|_{1} ...
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Normed space and Subspace

Let $X$ be a vector space with norm and $Y$ subspace with non empty interior ($Y^\circ\neq\varnothing$), then $X=Y$. I'm trying to use the theorem that says: if $X,Y$ normed spaces and $T:X\to Y$ ...
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Saturation of the Babenko–Beckner inequality

The Babenko-Beckner inequality $|| \mathcal F f ||_q \geq C(q,p)||f||_p$ is a well-established theorem. It relates the $q$-Norm of a Fourier transform $\mathcal F f$ of a function $f$ to its ...
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24 views

How to prove that the infinity norm of a matrix is the max of row sum?

I know how to prove that the 1-norm of a matrix is the max of the column sum, but not sure how to prove that the inf-norm is the max of the row sum. Any suggestion? Thanks
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21 views

>>A. $X$ banch. >>B.M banach. >>C. $X/M$ banach.

I want to prove that any two will imply the other: A. $X$ banch. B.M banach. C. $X/M$ banach. I have proved A+B imply C by taking M closed in X. I am not understanding ...
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How to “compare” vectors?

I'm reading the definition of matrix norm in Golub & Van Loan and came across this "It is clear that the p-norm of matrix A is the p-norm of the largest vector obtained by applying A to a unit ...
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22 views

A connection between a matrix norm and a related matrix's largest eigen-value

I have been asked to prove that for $A\in M_n(\mathbb{C})$, with $||A||:=\sup_{x\in\mathbb{C}^n,|x|=1}|Ax|$, $$||A||=\sqrt{\lambda}$$ where $\lambda$ is the eigen value of largest modulus of $A^*A$. ...
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143 views

Estimating the sum of a series ($\ell^1$ norm) in terms of two weighted $\ell^2$ norms

Does there exist a $C > 0$ such that $$ \sum_{n \geq 1} a_n \leq C \left( \sum_{n \geq 1} 2^n a_n^2 \right)^{1/4} \left( \sum_{n \geq 1} 2^{-n} a_n^2 \right)^{1/4} $$ for all $a_n \geq 0$ with ...
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24 views

Prove the following Norm Inequality [on hold]

Show that $\forall x \in \mathbb{C}^n$ $$\|x\|_2 \leq \|x\|_1 \leq \sqrt n \|x\|_2$$
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Norm and Cauchy sequence.

If someone could please show me how to show that if $x_n$ is a Cauchy sequence then $x_n \over ||x_n||$ is a Cauchy sequence as well? Thanks, I hope I've been clear enough.
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1answer
29 views

Norm of Outer Product

Let $x \in \mathbb{R}^N$ and $ y\in \mathbb{R}^M$. Show that $\|xy^T\|_{\infty}=\|x\|_{\infty}\>\|y\|_1$ I've been able to show the following: $\|xy^T\|_{\infty}= \|xIy^T\|_{\infty} \le ...
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25 views

Bounded Operator and p-norm (more difficult than it seems).

Let $\mathbb{R}^k$ and $\mathbb{R}$ be real vector spaces (with the usual operations of addition and scalar multiplication in each one of them) with the norm $\|\mathbf{x}\|_p$ for the space ...
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49 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 ...
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31 views

Prove that $(\|x\|^p_X + \|y\|^p_Y)^{1/p}$ is a norm

Let $X$ and $Y$ be normed spaces equipped with the norms $\|\cdot\|_X$ and $\|\cdot\|_Y$, then prove that the following defines a norm on $X\times Y$ for $1\le p < \infty$: $\|(x,y)\| := ...
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1answer
20 views

Why $||y|| = \max_{||b|| \leq 1} y^tb$?

I read Application of Legendre transformation in computer vision And at part 5.1 i found strange equality $||y|| = \max_{||b|| \leq 1} y^tb$ Can anyone provide me intuition why this equality ...
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1answer
91 views

Reading a Laplacian Matrix and its labeled graph?

How can the following labeled graph be extracted from the Laplacian Matrix below and viceversa? I had a look at this great conversation but it is already too advanced for me.
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Transformations invariant wrt. $L_1$ norm.

$A$ is a real matrix of size $n \times k$, where $k \leq n$. $A$ has independent columns. Characterize the class of matrices $M \in \mathbb{R}^{k \times k}$ such that: $\forall x \in \mathbb{R}^k.\; ...
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Showing coercivity of a function

I am well attuned to the definition for a function to be coerce, which is that $\lim_{\|x\| \to \infty}f(x) = \infty$ ie the values of $f$ go to infinity as the norm goes to infinity. So Ex.1 ...
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1answer
34 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 ...
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1answer
17 views

Prove convolution $f\ast g\in L^\infty(\mathbb{R})$

Let $f\in L^p(\mathbb{R}),g\in L^q(\mathbb{R})$ ($1\le p,q<\infty:\frac 1 p+\frac 1 q=1$). Prove that $L^\infty(\mathbb{R}) \ni f\ast g$ (the convolution of them) and also prove that $$\Vert ...
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How to estimate norms involving $|a-b|$?

I know the title isn't the best one. Here's my problem: Whenever I'm given functionals such as: $$\phi: \mathcal{C}^1([0,1]) \ni f \rightarrow f(\frac{1}{3}) - f'(\frac{2}{3}) \in \mathbb{R}, \ \ ...
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1answer
957 views

Why does spectral norm equal the largest singular value?

This may be a trivial question yet I was unable to find an answer: $$\left \| A \right \| _2=\sqrt{\lambda_{\text{max}}(A^{^*}A)}=\sigma_{\text{max}}(A)$$ where the spectral norm $\left \| A \right ...
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15 views

Dual norm of induced norm on matrices

Given a norm on a (finite dimensional) vector space $V$, we can define the induced norm on linear maps from $V \rightarrow V$, by $\|M\|_{I(V)} = \sup\limits_{\|x\|=1}\|Mx\|$. It is also true that ...
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19 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 ...
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20 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 ...
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20 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 ...
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28 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 ...
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1answer
21 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, $e_1$ ...
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33 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) ...
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1answer
923 views

Matrix Norms Inequality Proof

How do I prove this inequality / equivalence about matrix p-norms? It appears on the wikipedia and mathworld.wolfram pages on matrix norms without proof. $\|A\|^2_2 \leq \|A\|_1 \|A|\|_\infty$ Maybe ...
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60 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 ...
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1answer
65 views

Why is $z^T x \leq \|x\| \| z \|_*$ for dual norm in $\mathbb{R}^n$?

This is probably very obvious, I was looking at this. It looks so much like a Cauchy-Schwarz though, and I would say it is very obvious from the definition if it wasn't for the condition that $\|x\| ...
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1answer
52 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 ...
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18 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.
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27 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 ...
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1answer
19 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 ...
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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 ...
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Chebyshev's Inequality and the length of a random vector

Suppose that we take $n$ iid random variables $X_1,\dotsc, X_n\sim \operatorname{Unif}[-1, 1] $ and define $Y_n=\lVert (X_1,\dotsc, X_n) ^\intercal \rVert $. By what means can we employ ...
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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 ...
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1answer
59 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 ...
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1answer
27 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: ...
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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| = ...
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61 views

Lipschitz continuity and gradient of a real-valued function on a normed space

The gradient of a function $f:E\to\mathbb{R}$ is Lipschitz continuous with parameter $L > 0$ iff $$\|\nabla f(x) - \nabla f(y)\|^* \le L\|x-y\| \quad \forall x,y\in E.$$ I have two questions: ...
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42 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 \| ...
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79 views

Inequality on matrix norm: $ \lVert A^n \rVert \leq \lVert A \rVert^n $ [closed]

If $A$ is a $n \times n$ matrix and assume we have a matrix norm $\lVert \cdot \rVert$. In a proof I need the following property: $ \lVert A^n \rVert \leq \lVert A \rVert^n $. I don't know how to ...
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1answer
33 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 ...
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1answer
29 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 ...
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2answers
53 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 $\alpha$.