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

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How to show that a function in $L_2$ space is uniformly bounded in the “truncated norm”

Let $$L_2[0, \infty) = \left\{f: \mathbb{R}_{+} \to \mathbb{R}^n \mid \int_0^\infty f(t)^Tf(t) \, dt< \infty \right\}$$ Define the truncated norm as $$\|f\|_K = \sqrt{\int_0^K f(t)^Tf(t) \, ...
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1answer
6 views

Expression related to dual norm on bounded linear functionals

Given a vector space $V$ and a norm $\|\cdot\|$ on $V$, the dual norm $\|\cdot\|^*$ on $V^*$ is given by $\|f\|^* = \sup \left\{\frac{f(v)}{\|v\|}\right\}$ over all nonzero vectors $v$. I've found ...
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22 views

Operator norm of a diagonal matrix

I want to prove that the operator norm of a diagonal matrix $D$ is less than or equal to its largest value. I've tried the following but I don't know if it is correct. ...
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1answer
31 views

Norm square of an integral

Is it allowable to do like this: $$ \Psi(x,t) = \int_{-\infty}^{+\infty} e^{k^2/a}*e^{ikx}*e^{-ik^2t} $$ $$ |\Psi(x,t)|^2 = |\int_{-\infty}^{+\infty} e^{k^2/a}*e^{-ikx}*e^{-ik^2t}dk|^2 $$ $$ ...
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1answer
57 views

$A$ and $B$ are positive matrices and $\|A+B\| = \|A\|+\|B\|$, then A and B have a common eigenvector

Suppose $A$ and $B$ are positive matrices such that $\|A+B\| = \|A\|+\|B\|$. Show that $A$ and $B$ have a common eigenvector, where $\|A\|$ is the operator norm of $A$. I'm wondering if there is a ...
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1answer
106 views

Are the norms $\Vert f\Vert_1=\Vert f\Vert _\infty+\Vert f'\Vert _\infty$ and $\Vert f\Vert _2=\vert f(a)\vert +\Vert f'\Vert _\infty$ equivalent?

We have the norms $\Vert f\Vert_1=\Vert f\Vert _\infty+\Vert f'\Vert _\infty$ and $\Vert f\Vert _2=\vert f(a)\vert +\Vert f'\Vert _\infty$ where $f\in C^1[a,b]$. Are they equivalent and how shoud I ...
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Proof the inequalitiy for the two Matrices $A, B$

Let $ A,B \in C^{nxn}$ then $$ 1 \le || (\lambda I-B)^{-1})(A-B)||\le ||(\lambda I-B)^{-1}||*||(A-B)||$$ for any eigenvalue $\lambda $ of $ A $ which is not an eigenvalue of $ B$ and any operator ...
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0answers
19 views

Matrix Norm Confusion

I am looking at my textbook which considers an example but I am not sure how it derived the matrix norm with $||A|| = \sqrt{9/2 + (1/2)\sqrt{65}}$ and was hoping someone could provide the calculations ...
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1answer
16 views

Frobenius norm of matrix $A^{T}A$ is $trace(A^{T}A)$? Where all of the values in the matrix $A$ are real

We know that frobenius norm of a matrix $A$ is given by $\|A\|_{F}=\sqrt{trace(A^{T}A)}$. Can we write frobenius norm of matrix $A^{T}A$ to be $\|A^{T}A\|_{F}=trace(A^{T}A)$, that is I am effectively ...
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For a normal operator is it true that $\|T^*T^2\| = \|T^3\|$?

For a normal operator is it always true that $\|T^*T^2\| = \|T^3\|$? See the accepted answer for the case in a Hilbert space Update: how about $\|T^*T^2\| = \|T\|^3$ in a Hilbert space
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1answer
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How to describe minimization of L1 norm error using linear programming?

Given a set of $n$ pair points $(x_1, y_1), ..., (x_n, y_n)$ in the plane, I need to find a line $ax + by = c$ that fits the points of the L1 norm error points as closely as possible. I need a linear ...
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1answer
138 views

log norm inequality for lower triangular part of matrix

Suppose $L$ is the lower triangular part of a matrix $A \in \mathbb{C}^{n\times n}$. Prove that $||L||_2 \leq ||A||_2 \log_2(2n)$. Here $||\cdot||_2$ is the matrix norm induced by the $p=2$ vector ...
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0answers
15 views
+50

Equivalences for a Banach lattice

I'm trying to prove the following equivalences for a Banach lattice $E$: $E$ has an order continuous norm Every monotone order bounded sequence in $E$ is convergent E is an ideal in ...
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0answers
39 views

What is the norm of the dual space $H^1(\Omega)'$?

I am working on the Bidomain-Model which, during a time interval [0,T], describes the electrical behaviour of the myocardial muscle considered as $\Omega \subset \mathbb{R}^3$. This model has partial ...
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2answers
104 views

About Cauchy–Schwarz inequality

For the vectors $x$ and $y$, the Cauchy–Schwarz inequality reads $$ |x\cdot y|\leq||x||\cdot||y|| $$ Does this inequality only hold for 2-norm? Or for any norms? Thanks in advance.
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23 views

Trace 0 and Norm 1 elements in Finite fields

Let $F_q$ and $F_{q^{\ell}}$ be the finite fields with $q, q^{\ell}$ elements respectively, where $\ell \ge 3$ is a prime and $\gcd(\ell, q)=1$. I have the following question: Does there exist ...
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0answers
12 views

Norm vs A-norm in non-Archimedean Functional Analysis

Let $K =(K,| \cdot |)$ be a non-Archimedean valued field. Let $E$ be a $K$-vector space. A norm on $E$ is a map $||\cdot||:E\to[0,\infty)$ such that: $||x||=0$ if and only if $x=0$, $||\lambda ...
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1answer
46 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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3answers
45 views

Show $Px \perp (x-Px)$ and prove $||P|| \leq 1$

Let $H$ be a Hilbert space and linear operator $P:H \rightarrow H$ satisfy $\langle Px,y\rangle = \langle x,Py\rangle$ for all $x,y \in H$ and $P^2=P$. (a) Show that $Px \perp (x-Px)$ for every $x ...
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1answer
19 views

Are $L_\infty$ functions measurable/integrable?

Lemma 2.6 of "Ergodic Theory with a view towards Number Theory" (Einsiedler-Ward) involves: $$ \int f d\mu $$ where $f \in L^{\infty}$. Actually it is a calligraphic $L$ and I'd love if you would ...
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Tighter upper bounds with ratios of powers of norms

This question arises in concentration or sparsity measures for finite sequences. Given $x\in \mathbb{R}^K$ and $1 \le r < s$, I try to find a tight upper bound for $$\psi_{r,s}(x) = \frac{\sum_1^K ...
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2answers
148 views

How does one prove $\left\lVert\sum\limits_{n=1}^{\infty}f_n\right\rVert\le \sum\limits_{n=1}^{\infty}\lVert f_n\rVert$?

Can the triangle inequality for norms works for infinite some as well? Hope that someone can give some hints to prove it or give a counterexample. Thanks!
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3answers
827 views

Show that the sup-norm is not derived from an inner product

I am trying to show that the norm $$\lVert{\cdot} \rVert _{\infty}=\sup_{t \in R}|x(t)|$$ does not come from an inner product (the norm is defined on all bounded and continuous real valued ...
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5answers
936 views

Is closure of convex subset of $X$ is again a convex subset of $X$?

Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...
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0answers
20 views

How to compute the gradient of the following matrix function? [on hold]

$f(X)=\left\|XX^T-I \right\|_F^2$, where $\left\|\cdot \right\|_F^2$ is Frobenius matrix norm
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288 views

Optimization of Frobenius Norm and Nuclear Norm

How to solve the following optimization problem, \begin{equation} \boldsymbol{\hat{x}} = argmin_{\boldsymbol{x}} \frac{1}{2} \| \boldsymbol{x - y} \|_F^2 + \lambda \| \boldsymbol{x} \|_{*} ...
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1answer
25 views

Where does this inequality come from: $\frac{|f(x) - f(x-h)|}{h} \le ||f'||_{L^\infty(x, x-1)}$?

I came across this inequality today: $$\frac{|f(x) - f(x-h)|}{h} \le ||f'||_{L^\infty(x, x-1)}$$ I realise if we let $h \to 0$ we obtain the derivative on the left hand side so I can see it has ...
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0answers
21 views

A confusion about the norm of the restriction of a linear mapping.

Let $\Bbb X$ be a Banach space, $T:\Bbb X\to \Bbb X$ be a linear map and $P:\Bbb X\to \Bbb X$ be a projection operator. Denote the closed subspace that is the range of $P$ by $\Bbb Y:=\mathcal R(P)$. ...
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1answer
37 views

Unique, minimal norm solution of a linear equation

Suppose an equation $Ax=b$ has non-unique solutions. Prove that there exist a unique vector $x_{min}$ satisfying $Ax_{min}=b$ whose norm is the smallest among the solutions of that equation. The ...
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1answer
30 views

Showing a certain map is a norm.

Define $$\|x\|=\sqrt[3]{(|x_1|^2+|x_2|^2)^{3/2}+|x_3|^3}$$ for any $x=(x_1,x_2,x_3)\in \Bbb R^3$. Show that $\|\cdot\|$ is a norm on $\Bbb R^3$. I stuck on showing triangle inequality. I don't know ...
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1answer
5k views

How to prove triangle inequality for $p$-norm?

If $\mathcal{M}=\{M_i : i\in I_n\}$ is a collection of metric spaces, each with metric $d_i$, we can make $M=\prod_{i\in I_n}M_i$ a metric space using the $p$-norm, we simply set $d : M\times M\to ...
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0answers
29 views

Finding $\|f\|$

Given $f: \ell^2 \rightarrow \mathbb R$ defined by $$f(a_1,a_2,...)=\sum_{n=1}^{\infty} \frac{(-1)^n}{3^{n-1}} a_n$$ Is $f$ a bounded linear functional? If yes, then find $\|f\|$. It is definitely ...
4
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1answer
252 views

Equivalents norms in Sobolev Spaces

I know that this is classical but I have never do the calculations to show that the norms in the sobolev space $W^{k,p}(\Omega)$ \begin{equation} \|u\|_{k,p,\Omega}= \Bigl(\int_{\Omega} ...
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1answer
39 views

Symmetric norms on $\mathbb{R}^n$ are between the $\ell^1$ and $\ell^\infty$ norms

We say that a norm $\|\cdot\|$ on $\mathbb{R}^n$ is symmetric if it is invariant under sign changes and permutations of the components. Suppose that $$\|(1, 0, ..., 0)\| = 1$$ for a symmetric norm. ...
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1answer
29 views

Topology in the set of matrices

Let $M_n(\mathbb{R})$ be the set of real $n\times n$ matrices. I've proved that the map $\left \|\cdot \right \| \mapsto \left \| A \right \| :=\sqrt{\text{tr}(A^tA)}$ is a norm. Then I defined the ...
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1answer
43 views

Some inequalities between 1- norm, 2- norm and infinity-norm: $\|x\|_2\leq\sqrt{\|x\|_1\| x\|_\infty}\leq\frac{1+\sqrt{n}}{2}\|x\|_2$

Let $x\in\mathbb{C}^n$. Do the following inequalities hold? $$\lVert x\lVert_2\leq\sqrt{\lVert x\lVert_1\lVert x\lVert_\infty}\leq\frac{1+\sqrt{n}}{2}\lVert x\lVert_2.$$ I think the first inequality ...
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0answers
7 views

Doubt on inferring relations between linearly transformed variables based on their norm inequalities

Consider two vectors $ x, y \in \mathbb{R}^n $ such that $ || x ||_p \le || y ||_p $ then, given an arbitrary matrix $ A \in \mathbb{R}^{m \times n} $ is $ || A x ||_p \le || A y ||_p $ a true ...
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1answer
18 views

Is there a way to compare the following two operator norms?

Suppose that $K:C([0,1])\to C([0,1])$ is a continuous operator both with respect to $L^2$ and $L^\infty$ norms. Consider the following operator norm $$\sup_{\|f\|_2\leq 1}\|Kf\|_\infty$$ where ...
2
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0answers
21 views

Find value of $p$ such that $\sum |b_n|^p $ converges

Let $(X , \|\cdot \|)=(\ell^2, \|\cdot \|_2)$. Let $e_n \in \ell^2$. Fora bounded linear functional $\phi$ on $X$, let $b_n=\phi (e_n)$ for each $n \geq 1$. Find the value of $p \geq 1$ such that the ...
2
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2answers
43 views

Linear functional is continuous $\implies$ it is bounded

Let $f:X \rightarrow \mathbb R$ be a continuous linear functional. Prove $f$ is bounded. Since it is continuous, $\forall \varepsilon >0$, there exists $\delta >)$ such that ...
2
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1answer
46 views
3
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1answer
116 views

Derivative of Frobenius norm of matrix logarithm with respect to scalar

I am stuck on finding $t$ such that: $\frac{\partial}{\partial t}\|\log_m(M\Lambda^tM^T)\|_F=0$, where $M$ is $n\times n$ positive definite matrix (not symmetric, not unitary), $\Lambda$ is $n\times ...
2
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2answers
45 views

Prove that $\max_{|z| = 1} |P(z)| \ge 1$

I got stuck on this problem: Given a polynomial on complex plane $P(z) = z^n + a_{n-1}z^{n-1} + ... + a_1 z + a_0$ for $z \in \mathbb{C}$. Prove that $\max_{|z| = 1} |P(z)| \ge 1$ What I tried ...
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9 views

$ \mathcal L_1 $-property inherited through normalization?

simple question regarding functional spaces. Assume $ u(t) $ is a signal defined for $ t \geq 0 $, with $ |u| \notin \mathcal L_1 $, and construct the signal $$ g(t) = \dfrac{|u(t)|}{\sqrt{1 + ...
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1answer
17 views

$L^p$-norm minimization under linear constraints: Does the optimum depend on $p$?

Consider the following norm minimization program: \begin{align} \label{1} &\min_{x \in \mathbb{R}^d} &&\lVert x - x_0 \rVert_p^p &(1)\\ &\text{subject to } &&Ax-b \ge 0 ...
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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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0answers
24 views

L2 Norm: Unfamiliar notation

In this article that I am reading, I am given a non-negative spectral function $w(\lambda)$ which is "interpreted as a weight function determining the scalar product of two functions $f(\lambda)$ and ...
3
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1answer
268 views

Any positive linear functional $\phi$ on $\ell^\infty$ is a bounded linear operator and has $\|\phi \| = \phi((1,1,…)) $

This is a small exercise that I just can't seem to figure out. When I see it I'll probably go 'ahhh!', but so far I haven't made any progress. I'd like to prove that any linear functional $\phi$ on ...
3
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0answers
10 views

Exponentially weighted function in $ \mathcal L_1 $?

I have an interesting adaptive control problem. Consider a signal $u(t)$ generated by normalizing another signal, so that $$ 0 \leq u(t) < 1. $$ Consider the function generated from $u(t)$ as ...