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

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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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49 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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26 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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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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55 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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23 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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149 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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60 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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86 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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67 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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26 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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75 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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58 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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25 views

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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74 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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91 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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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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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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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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$x \in \mathbb R^d$, $B$(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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21 views

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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119 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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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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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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93 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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45 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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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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>>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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370 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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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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31 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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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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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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46 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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25 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 a strange equality $||y|| = \max_{||b|| \leq 1} y^tb$ Can anyone provide me intuition why this equality is ...
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107 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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26 views

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 ...