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

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Calculating the norm of an exterior product

I am trying to figure out how to calculate this quantity: $$ \frac{\lVert U^{t}_{\mathbf{x_0}}\mathbf{e}_1\wedge U^{t}_{\mathbf{x_0}}\mathbf{e}_2\wedge\ldots\wedge ...
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38 views

Given justification for induced norm for nonsingular matrix

Help!! Determine whether the following statement is true or false and give a justifcation: If A is nonsingular, then for any induced norm, $$ ||A^{-1}|| = ||A||^{-1}$$ Thanks!
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23 views

$\ell_1$ and $\ell_2$ norms on $\mathbb{R}^n$

I've been working on proving the following for a homework problem: Let $x \in \mathbb{ R } ^n$ and consider the $\ell_1$ and $\ell_2$ vector norms defined by $$ \| x \|_1 = \sum_{i=1}^n |x_i| \ , \ ...
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26 views

How does the inner product (-,-) with norm ||.|| relate to bilinear and linear forms in the FEM?

I am attempting to prove the Lax-Milgram lemma for the weak formulation of the finite element method. However I first need to prove continuity of the bilinear and linear forms ($a(u,v)$ and $l(v)$), ...
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Show that $ \|A \|_1= \|A^H \|_ \infty$ [closed]

With the definition of norm-1 and norm-infinity on matrice $ A_{m*n} $ that is: $$ \|A \|_1=\max_j \sum_{i=1}^ m |a_{ij}|$$ and $$ \|A \|_ \infty = \max_i \sum_{j=1}^ n |a_{ij}|$$ prove that: ...
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64 views

Show that $ \lVert A \rVert_2^2 \leq \lVert A \rVert _1 \lVert A \rVert _ \infty $

With the definition of $ \lVert A \rVert_2$ and $\lVert A \rVert_1$ and $\lVert A \rVert_ \infty$ that is: \begin{gather} \lVert A\rVert_1 = \max_{j} \sum_{i=1}^m \lvert a_{ij}\rvert\\ \lVert ...
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45 views

Matrix and Vector Norm with diagonal matrices

I don't even know where to begin with a problem like this. Where should I start? A thorough (no shortcuts )answer and explanation is greatly appreciated. Let A be symmetric positive definite. Show ...
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46 views

Convexity of a certain function connected to the norm

Suppose that we are given two vectors $x,y$ in a normed space $X$. Can we prove in general that the function $$t\mapsto \|x-ty\|$$ is convex? It is certainly the case if the normed space has ...
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17 views

Piecewise absolutely continuous functions: norm implication

I have the following doubt: Consider a sequence of functions $(f_k)$, $k=1,2,\ldots$ which are piecewise absolutely continuous functions $f_k: [a,b] \rightarrow \mathcal{R}^n$ with a finite number of ...
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28 views

Operators on $l^{\infty}$

Could you tell me how to find norms of these maps? $$ \phi : l^ {\infty} \ni \{x_n \} \rightarrow \left\{ \frac{x_n}{n} \right\} \in l^2$$ and $$\phi : l^ {\infty} \ni \{x_n \} \rightarrow \left\{ ...
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14 views

Norm of a functional given by difference of values

How to calculate the norm of such functional? $$\phi : \mathcal{C} ([0,1]) \ni f \rightarrow 3 f (\frac{1}{2}) - 5 f (\frac{2}{3}) \in \mathbb{K} (= \mathbb{R}, \mathbb{C})$$ If we equip ...
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1answer
20 views

Norm of a Vector equality

As I prepare for the exam, I have encountered the following question: I am not very good on this Norm calculations, so your help is important. This is where I am so far: How do I continue from ...
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1answer
47 views

Bounded functions, existence of a functional with a given norm

Let $\mathcal{B} ([0,1])$ be the space of all bounded real functions on $[0,1]$ with sup norm. Prove that for each $x_0 \in [0,1]$ there exists a functional $\varphi \in \mathcal{B} ([0,1])'$, $|| ...
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23 views

Operator on continuous functions, its norm is not attained

Prove that the norm of this linear functional : $$\phi: \mathcal{C} ([0,1]) \ni f \rightarrow \int_0 ^{1/2} f(t)dt - \int_{1/2}^1 f(t)dt \in \mathbb{K} $$ equals $1$ if $\mathcal{C} ([0,1])$ is ...
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129 views

Norm of a linear map is not attained

Prove that the norm of the linear functional $$\phi: l^1 \ni \{x_n \} \rightarrow \sum_{n=1} ^{\infty} (1 - \frac{1}{n} )x_n \in \mathbb{K}$$ equals one but there doesn't exist a sequence $ \{x_n \} ...
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51 views

Norm of a linear map, rational numbers, continuous functions

Let $\{ x_n \}$ be an injective sequence of all rational numbers in $[0,1]$. Let $\mathcal{C} ([0,1])$ with complex values be equipped with the norm: $$||f|| = \sqrt{\sum_{n=0}^{\infty} 2^{-n} ...
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19 views

Operator on polynomials, antiderivative

We are given a linear map: $$\mathbb{R} [X] \ni p \rightarrow q \in \mathbb{R} [X], \ \ q'=p, \ \ q(0)=0 $$ and two norms on $ \mathbb{R} [X]$ : $||p||_{\infty} = \sup _{t \in [0,1]} |p(t)|$, ...
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120 views

Sub-gradient of the “$\ell_0$ norm”

I am trying to characterize the sub-gradient of l0-norm ($f(x) = ||x||_0=\sum_{i=1}^n 1\{{x_i \neq 0}\}$). At first, I thought l0-norm is a convex non-smooth function since it satisfies the triangle ...
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26 views

Relation between norms

if A is an $m\times n$ matrix prove that: $$\frac{1}{\sqrt{m}} \|A\|_1 \leq \|A\|_2 \leq \sqrt{n} \|A\|_1 $$
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What is $\sum_{i=1}^m \frac{1}{a_i^Tx-b_i}$

Ask a dumb question: We all know the following: $\sum_{i=1}^m |a_i^Tx-b_i| = ||Ax-b||_1$ How about $\sum_{i=1}^m \frac{1}{a_i^Tx-b_i}$? I think it is definitely not $ ...
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48 views

Convergence in norm and in mean

Here is my problem: If $\{f_k\}$ is a sequence in $L^2$ and $f_k\to f$ in mean, show that $\{||f_k||_2\}$ is a bounded sequence of real numbers. Before I start doing the problem, I would like to know ...
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78 views

Is $\|\nabla u\|_2$ a norm on $W^{1,2}(\mathbb{R}^N)$

Is $\|\nabla u\|_2$ a norm on $W^{1,2}(\mathbb{R}^N)$? I know when $\Omega$ is bounded, $\|\nabla u \|_2$ defines a equivalent norm on $W_0^{1,2}(\Omega)$ with $\|u\|_{1,2}$. And to me it seems ...
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66 views

How do I find the norm of a matrix?

I have the following matrix below and I would like to find the norm of the matrix. I learned that the norm of a matrix is the square root of the maximum eigenvalue multiplied by the transpose of the ...
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21 views

Equality between 2-Norm of polynomials

Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+ \dots a_0$ be a polynomial of $\mathbb{C}[x]$, we define $$||f||_2= \sqrt{ \sum_i|a_i|^2}$$ Now, studying Computational Algebra, I've read about the following ...
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Using Lagrange's Mean Value Theorem to prove equality of norms

I'm looking for a proof using Lagrange's Mean Value Theorem of the following: Prove that in $C^1[0,1]$ vector-space, $\left\|f\right\|=\left|f(0)\right|+\left\|f'\right\|_\infty$ norm is equivalent ...
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63 views

Does isomorphism between normed space implies equivalence of norm?

Let $E$ be a vector space, $\|\cdot\|_1,\|\cdot\|_2$ be two norms on $E$, if $T:(E,\|\cdot\|_1)\to (E,\|\cdot\|_2)$ be an isomorphism (linear bijection, $T,T^{-1}$ are bouned), then does this imply ...
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48 views

Vector norm in R^2

I want to create a vector norm in $R^2$ that is not identical to a p-norm. I'm thinking of slightly modifying the $p1$ norm, such as adding a coefficient or exponent. For example, $||x|| = ...
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Proving $||x||_{\infty}$ is a norm on $\mathbb{C}^n$

I'm trying to show that given $x,y \in \mathbb{C}^n$, the following holds $$ ||x + y||_{\infty} \leq ||x||_{\infty} + ||y||_{\infty} $$ Assume $x = a + ib, y = c + id, a,b,c,d \in \mathbb{R}$. $$ ...
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47 views

How do you express the Frobenius norm of a Matrix as the squared norm of its singular values?

Let the Frobenius norm of an m by n ($m \times n$) matrix M be: $$|| M ||_{F} = \sqrt{\sum_{i,j} M^2_{i,j}}$$ I was told that it can be proved that, if M can be expressed as follows (which we can ...
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Rank, nuclear norm and Frobenius norm of a matrix.

The nuclear norm, denoted $\|\cdot\|_*$ is a good surrogate for the $rank$ when minimizing problems like $$\label{pb1}\tag{1} \min_X rank(X) : AX = B $$ Here, we're trying to find an matrix X with low ...
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70 views

How to prove that 2-norm of matrix A is <= infinite norm of matrix A

Now a bit of a disclaimer, its been two years since I last took a math class, so I have little to no memory of how to construct or go about formulating proofs. My tutor unfortunately is very and ...
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35 views

Norm of Fredholm operator in $L^1$

Let $T:L^1([0,1])\rightarrow L^1([0,1])$ be the Fredholm integral operator given by $$ Tf(x)=\int_0^1 k(x,y)f(y)\, dy $$ where $k \in C([0,1]^2)$ is called the kernel of $T$. My problem is to find ...
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Given a p.s.d. matrix $X$ and vectors $a, b$, what is the relationship between $||a-b||_2$ and $a^TXa-b^TXb$?

Given $X \in \mathbb{R}^{N \times N}$, $a,b \in \mathbb{R}^N$ What is the relationship between $||a-b||_2$ and $a^TXa-b^TXb$?
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Example of infinite sequences that produces different norms

So I have an infinite sequence: $u = (u_1, u_2, u_3, . . .)$, where $u_j$ are real numbers. $l_1, l_2, l_\infty$ is defined as $$||u||_1=\sum_{j\in \mathbb{N}} |u_j|$$ ...
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32 views

Show that $\displaystyle |||A|||_{\infty}=\sup \limits_{1 \le i \le m} \sum_{j=1}^{n}|a_{ij}|$

Show that $$|||A|||_{\infty}=\sup_{||x||_{\infty}=1}||Ax||_{\infty}$$ for $A=[a_{ij}]$ an $m \times n$ matrix. Show that $$ |||A|||_{\infty}=\sup_{1 \le i \le m} \sum_{j=1}^{n}|a_{ij}|$$ Since the ...
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51 views

Proof for Norms in Vector Spaces

Prove that if a norm $\|x\|$ on a real vector space satisfies the parallelogram law, then the polarization identity defines an inner product and that the norm associated with this inner product is the ...
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39 views

$x \in \mathbb R^d$, $B$(closed)$ \subseteq \mathbb R^d$. Where $\mathbb R^d$ is an inner product space. Proof help [duplicate]

$x \in \mathbb R^d$, $B$(closed)$ \subseteq \mathbb R^d$. Where $\mathbb R^d$ is an inner product space. Show that there exists a point $b_0 \in B$ such that $d(x,B) = \|x - b_0\|$ My attempt: ...
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Different examples in L^p spaces

I was introduced to the concept of $L^p$ spaces for the first time, and has a lot of questions on what exactly they are. If I consider the measure space $(\mathbb{R},\mathscr{B}, Leb)$, for any $p, ...
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Norms and Banach Space

am nt understanding the meaning of H_T and how did we get ||(x,y)||_H|T
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72 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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Compute the Frobenius norm

I'm trying to compute the Frobenius norm of $L^{-1}$ where $$L^{-1}= I + N + N^2 + ... + N^{n-1}$$ and $N$ is strictly lower triangular. Can anyone suggest some way to do this?
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75 views

Normable topology determined by its restriction to a finite number of factors?

Is it generally true that all norms $\|\cdot\|$ on a finite product of normed spaces $E_1\times\dots\times E_n$ with $\|(0,\dots,0,x,0,\dots,0)\|=\|x\|_i$ where $\|\cdot\|_i$ denotes the norm on $E_i$ ...
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38 views

Vector Cross Product and Expression for perpendicular distance between any two Vectors

If $B \ne C$, prove that the perpendicular distance from $A$ to the line through $B$ and $C$ is $$\dfrac {|| (A-B)\times(C-B)||}{||B-C||} $$ where $\times$ means the vector cross product. Attempt: ...
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Limit to zero of the $p$-norm

I have the $p$-norm defined as $$\|x\|_p=\left(\sum_{k=1}^n|x_k|^p\right)^\frac{1}{p}.$$ I am trying to find the limit as $p\to0^+$ of $\|x\|_p$. I've seen it defined as $\{x_k:x_k\neq0\}$. Why is ...
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79 views

Matrix function to express pair-wise distances of rows in $X, Y$

There are two real matrices: $X, Y$ with $X$ being of dimension $n_1$ x $p$, $Y$ of dimension $n_2$ x $p$. The goal is to form the matrix $D$ of dimension $n_1$ x $n_2$ where each element $d_{ij}$ ...
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36 views

Operator norm of a matrix less than or equal to one

Do all matrices of operator norm $\leq 1$ have the sum of the absolute values of their rows $\leq 1$?
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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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9 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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18 views

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 ...