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

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Let $\alpha , \beta \in \mathbb{Z}[\sqrt{d}]$. Prove that $\alpha$ is a unit $\iff N(\alpha) = 1$. [duplicate]

Where $N(\alpha)$ is the norm of alpha. I have proved in a previous question that $N(\alpha\beta) = N(\alpha)N(\beta)$. and I have no trouble proving the $\Rightarrow$ but I don't know how to prove ...
2
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19 views

Norms under Conjugation by Projection Opertaros

I was reading about equivalent forms of the Kadison Singer Problem, and while looking at the Feichtinger Conjecture, I came across the claim that, for a projection operator $P$ and a self-adjoint ...
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1answer
40 views

Matrix induced by p-norm vector defintion

I'm having a bit of trouble understanding the exact definition of a matrix norm that is induced by the vector norm. In this specific case, our matrix norm definition is: $$||A|| = \max\limits_{x \neq ...
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1answer
50 views

Does elementwise matrix inequality extend to norms?

The elements of $A$ and $B$ are non-negative and $A_{ij} \leq B_{ij} \; \forall \; i,j$. Is it true that $\Vert A \Vert_p \leq \Vert B \Vert_p$ ? The norm is the operator norm induced by the usual ...
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1answer
48 views

What is the norm of this linear operator?

I need to show that $\|A(g_n)\|_1=b-a-\frac1n$ where $A:L^1([a,b])\to L^1([a,b])$ is given by $A(f)(x) = \displaystyle\int_a^xf(t)~dt$, and $g_n:[a,b]\to\mathbb R$ is given by $$g_n(t)= \begin{cases}n ...
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1answer
83 views

What makes a norm “appropriate”? Why can't testfunctions be normed appropriately

I often hear the term "using an appropriate norm". Then I once read that the $C^\infty_0(\Omega)$ cannot be appropriately normed. Why is that? Furthermore, when doing some numerical analysis you often ...
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1answer
51 views

Show that $\langle y_i, y_j\rangle = 0 \forall i \neq j.$

Let $y_1, y_2, . . .$ be a sequence in a Hilbert space. Let $V_n$ be the linear span of $\{y_1, y_2, . . . , y_n\}$. Assume that for $n \ge 1, ∥y_{n+1}∥ \le ∥y − y_{n+1}∥$ for each $y \in V_n$. ...
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2answers
39 views

minimum value of ||u-v||, given value of ||u|| and ||v||

given $\|u\|=2$ and $\|v\|=3$ Question: What is the minimum value of $\|u-v\|$? The || sign the the norm How do I go about solving this? $\|u-v\|\le\|u\|+\|v\|$?
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21 views

Norm of kronecker product

Given square matrices $A,B$. Does the equality $\|A\otimes B\|_p=\|A\|_p\|B\|_p$ hold? what if $p=2$? Show why.
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1answer
22 views

Squared reverse triangle inequality

Is there a possibility to obtain a bound on the difference of squared norm $\left\lvert\Vert x \Vert^2 - \Vert y \Vert^2\right\rvert$ in terms of the norm of the difference $\Vert x - y \Vert$? I ...
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1answer
58 views

Inequality between the norm of derivative and the derivative of norm

Let $x(t)=[x_1(t)~x_2(t)~\cdots ~x_n(t)]^T$, function $x_i:R\rightarrow R$ is differentiable, then it can be drawn that when $p=2$, $\|\frac{d}{dt}x(t)\|_p\geq \frac{d}{dt}\|x(t)\|_p$. I wonder if ...
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17 views

convex envelop on a unit ball

What does that exactly mean by "the function $g$ (continuous) is the convex envelop of the function $f$ (discrete) on a unit $\ell_\infty$ ball". I understand the part of "a function being the convex ...
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39 views

How to prove this inequation (About matrix and vector norm)? [closed]

$\|A\vec{x}\|\leq\|A\|\space\|\vec{x}\|$ where $A$ is a $m\times n$ matrix and $\vec{x}$ is a n-dimensional column vector. Assume that $\|A\|=\sqrt{\Sigma_{i}\Sigma_{j}a_{ij}^{2}}$
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17 views

Norm on following matrices

I know that for infinite, ||A|| = max(absolute values of added first row, second row, third row). But how do you find the for p = 2 and F?
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34 views

Lipschitz continuity of matrix product

Let $x\in\mathbb{R}^n$ and define the matrix functions $A:\mathbb{R}^n\to\mathbb{R}^{m\times p}$ and $B:\mathbb{R}^{n}\to\mathbb{R}^{p\times p}$. Define $F(x) = ...
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1answer
35 views

Equivalent norms without Cauchy-Schwarz inequality

Let $X$ be a finite-dimensional vector space over $\mathbb{F}$. ($\mathbb{R}$ or $\mathbb{C}$) Theorem: All norms on $X$ are equivalent. Proof: $a_k$s and $c_k$s will refer to elements of ...
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1answer
27 views

$L_p$ norm $\leq L_2$ norm for $1\leq p\leq2$ for Random Variables

Let {$X_i;i\geq0$} be a sequence of random variables defined on the probability space ($\Omega,\mathcal{F},P$). If $||.||_p$ is the $L^p$ norm defined as $||X_i||_p=(E[|X_i|^p])^{1/p}$, how should I ...
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21 views

Can I get the upper bound of the $L_2$ norm of a symmetric matrix?

Suppose there is a symmetric matrix ${\textbf{A}}_{d\times d}$ with each element $a_{ij}$ draw from a Gaussian distribution $\mathcal{N}(0,\delta)$. Is there an upper bound of ${\left\| \textbf{A} ...
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21 views

Converting a norm-computation SemiDefinite program to standard SDP form.

I'm trying to express this norm-computation semidefinite program: for given $A \in R^{m \times n}$ and a scalar $\epsilon \in (0,1)$ $$\gamma_{2}^{\epsilon}(A):= \min\,t\,\, subject\, to\, \left( ...
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38 views

Triangle inequality in norms

Let $N:V\to\mathbb{R}$ be a sequence of norms and $d(x,y)=\sum_{j=1}^{\infty}2^{-j}\frac{N_{j}(y-x)}{1+N_j(x)+N_j(y)}$ Show that $d(x,z)\leq{d(x,y)}+d(y,z)$ I tried the following ...
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1answer
34 views

$\sim$ is an equivalence relation on the set of norms on $X$.

Let $X$ be a vetor space. Prove that $\sim$ is an equivalence relation on the set of norms on $X$. Where $\sim $ is the equivalence of two norms. This seems very abstract. What exactly do I have to ...
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1answer
65 views

Prove $\|\cdot\|_p$ and $\|\cdot \|_q$ aren't equivalent on $\ell^p$

$1 \leq p < q < \infty$ So we need to find something in $\ell ^p$ that gives different results in each of the two norms but I can't think of anything. I could think of something that is n ...
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1answer
38 views

Prove $\| \cdot \|_{\infty}$ is well-defined on $\ell ^1$

$1 \leq p < q \leq \infty$ $p=1$ and $q= \infty$ Let $(a_k)_{k \geq 1} \in \ell^1$, then $$\sum_{k \geq 1}|a_k|^1< \infty$$ (*) which means $\sup_{k \geq 1 } |a_k|^1 < \infty$. Hence it is ...
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1answer
49 views

Prove $\sum \frac{\sin(n^2 t)}{n^2}$ converges

Let $(X,\|\cdot\|)$ be a Banach space and a sequence $(x_n) \subseteq X$ be such that the series $\sum_{n=1}^{\infty} \|x_n\|$ converges in $\mathbb R$. Prove that the sequence $S_n=x_1+\cdots+x_n$ ...
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1answer
51 views

Vector lengths: How to prove this inequality? [duplicate]

I plugged in the summations, tried to square both sides but couldn't reach the desired conclusion. I'm not sure what the second vector would be for the Cauchy inequality. Even when I use the vector ...
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1answer
30 views

Prove $\sup _{t \in [0,1]} |P(t)|$ is a norm

Let $X$ be the vector space containing all polynomials with real coefficients. For every $P \in X$, define $N_1 (P)=\sup _{t \in [0,1]} |P(t)|$ I am having trouble proving the part when you show ...
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24 views

Spectral radius of the product of a right stochastic matrix and hermitian matrix

Let us define the following matrix multiplication: $C=AB$ where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$ … $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M - \mu ...
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1answer
19 views

Spectral radius of the product of a right stochastic matrix and a block diagonal matrix

Let us define the following matrix multiplication: $C=AB$ where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$ … $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M - \mu ...
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2answers
77 views

Spectral radius of the product of two matrices

Let us define the following matrix multiplication: $C=AB$ where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$ … $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M - \mu ...
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1answer
44 views

Inequality between 1-norm and 2-norm of a vector in [on hold]

I have been trying for a while but still can't figure out how to prove this for vectors $x\in\mathbb R^n$: $$||x||_1 \leq \sqrt{n}||x||_2$$ I can prove the case when it's $\sqrt{n}||x||_\inf$ that's ...
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12 views

Quantifying the angle metric on the Grassmannian in terms of the norm on the exterior power

Let $V$ be a finite-dimensional Hilbert space and $G_k(V)$ the Grassmannian of $k$-dimensional subspaces of $V$. The $k$th exterior power $\bigwedge^k(V)$ can be equipped with a scalar product by ...
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Condition on definition of dual norm

In a review I'm reading, the dual norm is defined as $$||z||_*=\max_{||w||\leq 1}\langle w,z\rangle. $$ Though I'm having a hard time understanding why $||w||\leq 1$ isn't equivalent to $||w||=1$, ...
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1answer
30 views

Why are matrix norms defined the way they are?

Given $A$ a square matrix Define: $\|A\|_1$ as the max absolute column sum $\|A\|_2$ as the sum of the squares of each element $\|A\|_\infty$ as the max absolute row sum Pray tell, why are ...
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Prove the identity $\| R-A\| < \|Q-A\|$ for all $Q \in SO(3), Q\neq R$?

Let $A$ be a 3 by 3 matrix and positive-definite and let $A=RU$ be its right polar decomposition. How can I prove that $\| R-A\| < \|Q-A\|$ for all $Q \in SO(3), Q\neq R$? Note that $\|.\|$ is ...
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31 views

Find the operator norm

Let $T : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the linear transformation that the matrix $\begin{pmatrix} 1& a \\ 0 &1\end{pmatrix}$ determines. Find the operator norm $||T||$ with respect ...
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13 views

Is it possible to combine L2 norm optimization and thresholding?

Say you have a model like $g = x + \eta$, where $g$ is your observation vector, $x$ the unknown true vector and $\eta$ the additive noise vector. Now you intend to solve for $x$ by $min ||g-x||_2$. No ...
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1answer
56 views

Sequence of partial sums converge

Let $(X,\|.\|)$ be a Banach space and a sequence $(x_n) \subseteq X$ be such that the series $\sum_{n=1}^{\infty} \|x_n\|$ converges in $\mathbb R$. Prove that the sequence $S_n=x_1+...+x_n$ converges ...
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1answer
38 views

Proving these norms are not equivalent

$X$ is the vector space containting all polynomials with real coefficients. For every $P \in X$, define $N_1(P)= \sup _ {t \in [0,1]} |P(t)|$ and $N(P)=N_1(P)+|P'(1)|$. Prove that $N$ is not ...
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1answer
30 views

Norm of a real function

Given a function $g:\mathbb{R}^n\to\mathbb{R}$ which is nonnegative, suppose we take any norm of this function. Is it true to say $$\Vert g(x)\Vert =\vert g(x) \vert = g(x)?$$ Additionally, If we ...
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55 views

Norm in finding local extrema for functional

In "The Calculus of Variations" by Bruce van Brunt, he says: Let $J:C^2[x_0,x_1]\to\mathbb{R}$ be a functional of the form $$J(y)=\int_{x_0}^{x_1}f(x,y,y^\prime)dx,$$ where $f$ is a function ...
2
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1answer
58 views

L2 Norm of Pseudo-Inverse Relation with Minimum Singular Value

Consider a matrix $A \in\mathbb R^{n\times m}$ with $n>m$. It has full column rank, i.e. $\operatorname{rank}(A)=m$. Its left pseudo-inverse is given by; $$A^{-1}_\text{left}=(A^TA)^{-1}A^T $$ ...
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1answer
56 views

$a_n=\int|f|^nd\mu$ then $\lim_{n\to\infty}a_{n+1}/a_n=||f||_\infty$

Let $(X,\mu)$ be a measure space with $\mu(X)=1$. Let $f$ be a measurable function such that $0<||f||_{\infty}< \infty$. Let $a_n=\int|f|^nd\mu$ Show that: (1) ...
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200 views

The 2-norm of the integral vs the integral of the 2-norm

I`m currently having some issues with a seemingly innocent problem. I would like to show that $$\Bigg|\Bigg|\int_\mathbb{R}\begin{pmatrix}A(x)\\B(x)\end{pmatrix}dx\Bigg|\Bigg|_2 \leq ...
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1answer
26 views

Show a bound on the error

Suppose $g \in C[a,b]$, and $M(b-a) < 1$, and let $y$, $y_N$ $\in C[a,b]$ be the unique solutions of the equations, $y = g + Ky$ and $y_N = g + K_N y_N$. $y_N$ has been previously defined as an ...
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1answer
42 views

prove absolute integrability given square integrability

am trying to follow the outline of a proof in a book i am reading - must be missing something obvious, but would like to understand what exactly... $f$ is complex and square integrable over e. g. [0, ...
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43 views

How to see if a norm is finite

How do you know if $$||\frac{1}{\sqrt3},\frac{1}{\sqrt8}, ... , \frac{1}{\sqrt{n^2-1}}, ... ||_2$$ is finite. So this means is $$\bigg( \sum _{n=2}^{\infty} | \frac{1}{\sqrt{n^2-1}}|^2 \bigg) ...
4
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1answer
41 views

Bounding a function of norms on the unit cube

For a vector $v \in [0,1]^n$ and $p > 1$ we denote the p-norm of $v$ as: $||v||_p = (\sum_iv_i^p)^{\frac{1}{p}}$. where $v_i$ are the entries of $v$. Define the following (weird looking) function ...
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1answer
16 views

Non convex objective in SVM

In the formulation of svm.. The line underline says the norm of the vector w is a non convex constraint.. But how is this so.. Isn't norm a convex function.. Also aren't the other objectives ...
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1answer
32 views

2-Norm of Non-Square Matrices

So, the 2-norm of an m x n matrix for m=>n is defined by the max singular value/square of the max eigenvalue. But, if it's not square, and you're only given a matrix A (no x-vector), what do you do if ...
0
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1answer
42 views

Proving polynomial v.s. is a norm

Let $X$ be the vector space containing all polynomials with real coefficients. For every $P ∈ X$, define $N_1(P) = \sup_{t∈[0,1]} |P(t)|$ and $N(P) = N_1(P) + |P'(1)|$. I have to prove $N_1$ is a norm ...