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

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Prove or disprove: If $1=||A||>||B||$, then $A-B$ is nonsingular.

Prove or disprove: If $1=\|A\|>\|B\|$, then $A-B$ is nonsingular. I think that since $\|A-B\|>0$ by the given conditions we know it is nonsingular. Any solutions or hints are greatly ...
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27 views

Matrix norm inequality proof - does this use Cauchy-Schwarz?

The matrix norm for $A : \mathbb{R}^n \rightarrow \mathbb{R}^m$ (so $A$ is an $m \times n$ matrix) is given by $$\|A\| = \sup_{X \in \mathbb{R}^n \setminus \{0\}} \frac{|AX|}{|X|}$$ where $| \cdot |$ ...
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How to show that a function in $L_2$ space is uniformly bounded in the “truncated norm” [closed]

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 $$\|f\|_K = \sqrt{\int_0^K f(t)^Tf(t) \, dt}$$...
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14 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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27 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. $$D=\text{diag}(d_1,d_2,...,...
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1answer
45 views

Norm square of an integral

Is it allowable to do like this: $$ \Psi(x,t) = \int_{-\infty}^{+\infty} e^{k^2/a}*e^{ikx}*e^{-ik^2t} $$ $$ |\Psi(x,t)|^2 = |\int_{-\infty}^{+\infty} e^{k^2/a}*e^{-ikx}*e^{-ik^2t}dk|^2 $$ $$ |\Psi(x,...
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1answer
70 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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2answers
23 views

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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23 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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50 views

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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0answers
20 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 x||=...
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1answer
34 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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0answers
79 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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34 views

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 $E^{**}$...
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1answer
27 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
22 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
68 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
32 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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0answers
33 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 ...
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32 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
52 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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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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0answers
22 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 ...
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1answer
19 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 $\|.\|...
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1answer
50 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
10 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 + u^2(t)}}...
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1answer
25 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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2answers
57 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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32 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 $...
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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 ...
3
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1answer
71 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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1answer
32 views

Solving Norm-Constrained Homogeneous Linear Least Squares

I am learning how to solve a norm-constrained homogeneous linear least squares problem. min $(norm(Ax))^2$ for x such that norm(x) = 1 The problem is set up with a Lagrangian as follows: cost = $(...
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27 views

Showing that common p-Norm isn't a norm anymore for $0\lt p\lt 1$

I have the following problem: I need to show that the common p-Norm defined as: $$||.||_p: (v_1, \dots ,v_n) \to (\sum^n_{i=i} |v_i|^p)^{1/p}$$ doesn't constitute a norm on $\Bbb R^n$ for $n \ge 2$ ...
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2answers
43 views

p-Norm with p $\to$ infinity

I have to show that: for all vectors $v\in \Bbb R^n$: $\lim_{p\to \infty}||v||_p = max_{1\le i \le n}|v_i|$ with the $||.||_p$ defined as $$ ||.||_p: (v_1, \dots ,v_n) \to (\sum^n_{i=i} |v_i|^p)^{...
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1answer
27 views

A complex unital algebra which is a Banach space is also a Banach algebra

In my functional analysis class I got stuck on this question on Banach algebras: Let $ \mathbb{A} $ be a complex unital algebra (it has a unit i.e. a multiplicative identity) and $ \mathbb{A} $ is ...
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1answer
66 views

Matrix norm inequality : $\| Ax\| \leq |\lambda| \|x\|$, proof verification

Suppose that $A$ is a normal $n\times n$ matrix. Show that $\|Ax \| \geq |\lambda_n|\|x\|$ for all $x \in \mathbf{C}^n$, if $\lambda_n$ is the eigenvalue to $A$ of smallest absolute value. Is this ...
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1answer
22 views

Show $c_{00}$ is not closed under supremum norm

Show $Y=c_{00}$ is not closed under $(\ell^{\infty}, \|\cdot\|_{\infty})$. I know that I need to find a $(y_n) \in c_{00}$ such that this converges to $y$ with $y \notin c_{00}$. So we need $\|y_n -...
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0answers
18 views

Finding norms on a piecewise function

For each $n=1,2,...$ let the function $g_n \in C[0,1]$ be defined by \begin{equation} g_n(t)=\begin{cases} 2nt & 0 \leq t \leq 1/2n \\ 2-2nt & 1/2n \leq t \leq 1/n \\ 0 & 1/n \leq t \leq 1\...
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2answers
52 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 $|f(x)-f(y)|=|f(x-...
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0answers
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Proving linear operator is bounded

Prove that the formula $T(b_1,b_2,b_3,...,b_n,...) = (b_1, b_2/2 ,..., b_n/n ,...)$ defines a bounded linear operator $T : (ℓ^∞,∥·∥_∞)→(ℓ^∞,∥·∥_∞)$. Proving that it is linear is easy. Need help with ...
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31 views

Differential dimension of $W^{l,p}(\mathbb{R^n})$

I'm looking for the differential dimension of $W^{l,p}(\mathbb{R^n})$, defined as follows: Given a function space $Z(\mathbb{R^n})$, we say that $\mu\in \mathbb{R}$ is the differential dimension of $...
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1answer
33 views

What are the differences between $l^p$ space and $L^p$ space?

I always thought small $l^p$ space, or so-called Banach space, is the space equipped with a norm similar to vector norm as $$||x||_p = \left( \sum_{i\in\mathbb{N}} |x_i|^p \right)^{\frac{1}{p}}$$ ...
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Does a contraction converging in power series necessarily lead to the operator being a proper contraction?

I was recently met with this in my functional analysis class on which I am stuck: Let $ \mathbb{H} $ be a Hilbert space and let T be a contraction operator on $ \mathbb{H} $ (meaning $ ||T|| \...
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5answers
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Verify that $(a^2 + b^2)(c^2 + d^2)$ = $(ac - bd)^2 + (ad + bc)^2$ for any integer $a$,$b$,$c$,$d$

Part 1 - Verify that $(a^2 + b^2)(c^2 + d^2)$ = $(ac - bd)^2 + (ad + bc)^2$ for any integer $a$,$b$,$c$,$d$ Part 2 - Write 25988 as the sum of the two squares (of integers). A bit confused with ...
2
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2answers
74 views

Prove $f(x)=\|x\|$ differentiable everywhere but in $\{0\}$

I have the function $f: \mathbb R ^n \to \mathbb R$ where $f(x)=\|x\|$. I have to prove that $f$ is differentiable on $E$, where $E=\mathbb R^n \setminus \{0\} $, and show its derivative (for $x \ne 0 ...
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0answers
26 views

Find $\|B\|_{\ell^1 \rightarrow \ell^1} $

$B(a_1,a_2,...)=(a_1/1,a_2/2,\ldots, a_n/n,\ldots) $ This is a linear operator defined on $\ell^1$. Notice that $\|Bx\|=\sum |a_n/n| \le \sum |a_n| =\|x\|$. So $\|B\|=\sup_{\|x\|_1 \le 1} \|Bx\|...
5
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1answer
83 views

Does proper contraction on Hilbert space necessarily lead to convergence in norm to zero?

I was asked this in functional analysis class: Let $ \mathbb{H} $ be a Hilbert space and we are given an operator T satisfying: $ || Th || < ||h|| $ for all $ h \in H $. We are asked if ...
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1answer
31 views

Ratio of eigenvalues as the condition number of a matrix

I'm having an issue with a 2x2 matrix. My understanding is that one could use the ratio of the maximum eigenvalue to the minimum eigenvalue of a matrix in order to determine the condition number, ...
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0answers
21 views

Does strict contraction lead to convergence to zero in norm?

In my functional analysis class I was asked this question which got me stuck: Let $ \mathbb{H} $ be a Hilbert space and we are given an operator T satisfying: $ || T || < 1 $ in the ...
0
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1answer
20 views

Vector difference norm bound of $\|\frac{x_1}{\|x_1\|^2} - \frac{x_2}{\|x_2\|^2}\|$

Given $\|x_1 - x_2 \| \leq C$ where C is a constant, could we derive a bound of $\|\frac{x_1}{\|x_1\|^2} - \frac{x_2}{\|x_2\|^2}\|$?