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

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A norm inequality $(AB+BA') \leq |A+A'|*B \leq 2|A|*B$

Suppose that A and B are two $n \times n$ matrices. If $AB+BA'$ and $B$ are both positive definite symmetric matrices, is it true to conclude that $AB+BA' \leq $|A+A'|*B$ \leq 2|A|*B$? $A<B$ we ...
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If $||f-g|| < ||f^{-1}||^{-1}$, then $f$ is isomorphism implies $g$ is also isomorphism

Let $E$, $F$ be Banach space, $f,g \in L(E,F)$ and $f$ is isomorphism. Prove that if $||f-g|| < ||f^{-1}||^{-1}$, then $g$ is isomorphism. Hi everybody. I got stuck on this problem and can't ...
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Does this show the norm of this operator is zero?

We have $$T: C[-1,1]:\to \mathbb{R}$$ $$T(f)=\int_{-1}^1 x f(x) dx$$ The norm considered in $C[-1,1]$ is $$||f||=\max_{x\in[-1,1]} |f(x)|$$ So using $$||T||=\inf\{M:||Tf||\leq M||f||\}$$ in this ...
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Low-rank matrix approximation in terms of entry-wise $L_1$ norm

According to the Eckart–Young theorem, the low-rank matrix approximation problem $$\min_{\tilde{A}} \quad \| A - \tilde{A} \|_F \quad \text{s.t.} \quad \text{rank}(\tilde{A}) \le r$$ is given by the ...
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Calculate norm $1$ of $f(x)=2x^3+3x^5$ belonging to C[-1,1]

Calculate norm $1$ of $f(x)=2x^3+3x^5$ that belongs to $C[-1,1]$. As norm $1$ is called integral norm, I calculated the value of the function for the given interval, and the answer I get is zero. ...
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Minimize integral

Find numbers A and B such that the integral is minimal $$ \int_{0}^{\infty}\left\vert% \,\vphantom{\Large A}{\rm e}^{-x} - A{\rm e}^{-2x} - B{\rm e}^{-3x}\, \right\vert^{2}\,{\rm d}x $$ I have tried ...
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Is it true that $2^p\|f\|^p_{L_p} >2\|f\|^p_{L_\infty}$?

For any $C^1$ function defined in $(0,1)$, is it true that $$ 2^p\|f\|^p_{L_p} >2\|f\|^p_{L_\infty} $$ If it is true, how to prove it?
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In $C(\mathbb{R})$, the vector space over $\mathbb{R}$ of continuous functions from $\mathbb{R}$ to $\mathbb{R}$

In $C(\mathbb{R})$, the vector space over $\mathbb{R}$ of continuous functions from $\mathbb{R}$ to $\mathbb{R}$, let $W$ = { f ∈ $C(\mathbb{R})$ | $∫_{-∞}^∞$ | f(x) | $dx$ < ∞} where the ...
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Trouble in proving that $\|x\|_p = \max|x_j|$

We define p-norm in this way: $\|x\|_p = \{\sum ^N_j_=_1|x_j|^p\}^ {1\over p}$ We know that It change to $\|x\|_p = \max|x_j| $ when $ p \to \infty $ How can I prove this ?
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A basic doubt on the definition of induced matrix norm

In an optimization book I am following, the induced norm is defined as the maximum of the norms of the vectors $Ax$ where the vector $x$ runs over the set of all vectors with unit norm. Now, it says ...
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52 views

Equality with norms of matrices

I have a problem with prooving of following equality: $$\|E(I-\frac{ss^T}{s^Ts})\|_F^2=\|E\|_F^2-\frac{\|Es\|^2_2}{s^Ts},$$ where $E\in\mathbb{R}^{n\times n}$ and $0\neq s\in\mathbb{R}^n$. I tried to ...
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Why does a Norm Form have rational coefficients?

[From Wolfgang Schmidt's "Diophantine Approximation and Diophantine Equations"] "Let $K$ be a number field with $[K:\mathbb{Q}] = d$ and $L(X)$ be a linear form, say $L(X) = L(x_1,\dots,x_n) = ...
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On Real Hamilton Ring ..

i know the definition of real hamilton ring but if we said ,$ I$ is the ring of integral hamilton what does this mean ? what is the properites that word , integral , adds to the structure of ...
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225 views

Analysis.. Norm on C([a,b])

Let $w:[a,b]\rightarrow \mathbb{R}$ with $ w(x)\geq c>0 $ for some $c \in \mathbb{R}$ and all $x \in [a,b]$. Prove that $$\lVert f\rVert_w \ = \ \displaystyle\int^b_a \lvert f(t)\rvert w(t)\ ...
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36 views

A limit superior question in the context of the Neumann series

I'm trying to understand a step in the proof that the Neumann series converges: Let $X$ be a Banach space and $T\in L(X)$ (the space of bounded, i.e. continuous linear maps $X\to X$). It is known ...
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187 views

Inequality for singular value for differences of matrices (upper bound)

Does anybody know the inequality of singular value for differences of matrices, i.e. $\sigma_{max}\left(\begin{array}{c} A-B\end{array}\right)\leq??$ in term of $\sigma_{max}\left(\begin{array}{c} ...
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287 views

Is there any main method for finding norm of function in $L_1$ space?

Is there any main method for finding norm of function in $L_1$ space? For example : $f(x)$ = $\sin x$ in space $L_1[-\pi,\pi]$
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191 views

convergence in $L^2$

Let $ (f_n)$ be a sequence of measurable functions on $ [0,1]$ such that $\int_{0}^1 |f_n|^2 dm \le C$. Assume that there exist a function $f$ such that $\int_{0}^1 |f_n-f|dm \to 0$ as $n \to \infty ...
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How to do the following projection in Matlab?

I have 2 vectors $u$ and $v$ given in $\mathbb R^4$, e.g. $u = (-1,-2,3,4)$ and $v=(1,-2,-3,5)$ I also have $Ax=b$ which is an under-determined system; meaning, if $A$ is $m\times n$, then $m\le n$. ...
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What does RMSD mean?

Normally a rigid superposition which minimizes the RMSD is performed, and this minimum is returned. Given two sets of points and , the RMSD is defined as follows: $$\begin{align*} ...
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$l_1$ norm projection with regularization term

I recently encountered an optimization problem and looking for some technical paper for the same.The problem is give as below, $\min f(x)+\lambda*r(x) $ $\ s.t \ x \geq 0, ||x||_1 = 1$. where $x$ ...
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Vector and matrix norm definitions?

I've questions on these four norms whose definitions I'm memorizing like this: Vector euclidean norm: $(x_1^2+x_2^2+\cdots+x_n^2)^{1/2}$ Vector max norm: $\max\{|x_1|, |x_2|, \ldots, |x_n|\}$ Matrix ...
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What is the connection between strong norms and norms coming from scalar products (in pre-hilbert spaces)?

In the best-approximation problem of seperation theorems in convex analysis, there is the notion of a "strong norm", in the sense that If $\| x^1 + x^2 \| = \| x^1 \| + \|x^2 \| $, $x_1 , x_2 ...
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847 views

Linear algebra norm notation

I was reading a paper where the authors used the following notation: $||b - \mathbf{A}x||^2_D = (b - \mathbf{A}x)^t \mathbf{D} (b - \mathbf{A}x)$, where $\mathbf{D}$ is a diagonal matrix I was ...
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107 views

Orthogonal in the B Norm?

If you have two generalized eigenvectors $\varphi_1 , \varphi_2$ (with different eigenvalues) of a matrix A, then they will be orthogonal in the B norm. In this context, I do not ...
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Matrix norm characteristics

$$\|A\|_p = \displaystyle \max_{\|x\|_p = 1} \|Ax\|_p $$ $$\|A\|_2 \leq \|A\|_F \leq \sqrt{n}\|A\|_2$$ How I can show that $1$ and $2$ are correct? $2)$ $||Ax||_{2}=\sqrt{\sum_{i=1}^{n} ...
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175 views

parametrize hypersphere

I want to find an $n$-vector $\hat{\theta}$ that maximizes a function $f(\theta)$ subject to the $p$-norm constraint $||\theta||_p = c$. Is there a general parametrization of $p$-norm hyperspheres ...
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The linearness of extension of linear bounded operator

Let we have the following bounded linear operator $$T: D(T)\rightarrow Y$$ such that $D(T)$ is the domain and it is a vector space and $Y$ is a Banach space . Then it has an extension $$G: ...
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How to find the “two norm” of the difference between two vectors

I am using the Jacobi iterative method to estimate the solution to the system of equations $Ax=b$. With an iterative solver you take an initial (educated) guess as to what your $x$ may be, this is ...
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Relation between a function and its norm

While reading up on Sturm-Liouville system theory, I came across something I didn't fully understand. At one point, in the midst of proving the existence of solutions to the Sturm-Liouvill problem, ...
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21 views

Trace and Spectral norm of a matrix

Let $A_{n\times n}$ be a matrix. How I can show $$\vert \operatorname{trace} (A) \vert \leqslant n \sqrt{\rho(A^T A)}= n \Vert A \Vert_2$$ and if $A$ is symmetric and positive definite, ...
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33 views

Operator matrix norm associated with the vector norm

What is the operator matrix norm associated with this vector norm? $$\Vert x \Vert = \frac{1}{n} \mathop {\sum} \limits_{j=1}^{n} \vert x_j \vert, \qquad (x \in \mathbb{R^n})$$
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Prob. 14, Sec. 2.7 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the normed space of all ordered $n$-tuples of real numbers with the norm defined as ...
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Functional analysis, normed space

Can you please help me with this exercise.. I have to check if in interval $[a,b]$ continuously differentiable functions $x=x(t)$ norm can be defined as: $$\vert x(b) - x(a) \vert + \max_{a \leq t ...
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24 views

Gradient of this function

I am to solve an optimization problem as described below: $$ \min f(x) = \frac{1}{2}\left\lVert x - x_{b} \right\rVert^{2}+ \frac{1}{2}\left\lVert \epsilon \right\rVert^{2}$$ with $$ Hx -y = \epsilon ...
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28 views

Apply Cauchy-Schwartz to vector?

For $x_i, y_i \in \mathbb{C}$, the C-S inequality gives $$\left| \sum_{i=1}^n x_i \bar{y}_i \right|^2 \leq \sum_{j=1}^n |x_j|^2 \sum_{k=1}^n |y_k|^2 .$$ Is it true if $x_i, y_i$ are actually ...
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24 views

How is this norm $\|A\|_*$ called and what is it?

In my lecture notes I have found the notation $\|A\|_*$ for a matrix norm. Do you know the name of this norm (such that I can read the definition of it), or do you even know the definition of it? ...
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$\ell_1$ 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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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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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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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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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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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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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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27 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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21 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 strange equality $||y|| = \max_{||b|| \leq 1} y^tb$ Can anyone provide me intuition why this equality ...
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64 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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29 views

Prove convolution $f\ast g\in L^\infty(\mathbb{R})$

Let $f\in L^p(\mathbb{R}),g\in L^q(\mathbb{R})$ ($1\le p,q<\infty:\frac 1 p+\frac 1 q=1$). Prove that $L^\infty(\mathbb{R}) \ni f\ast g$ (the convolution of them) and also prove that $$\Vert ...
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36 views

Matrix norm properties: inequalities

Is the fololwing relationship always true: $x^{\top} (A - \frac{1}{2} \|A\|_F \, I_3) x > 0$, knowing that matrix A is definite positive?
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matrix F-norm inequalities on matrix sum and product

As you know, we have the two following inequalities for sum and product (for Frobenius norm): $\|A+B\|_F\leq\|A\|_F+\|B\|_F$ and $\|AB\|_F\leq\|A\|_F\|B\|_F$. The question is, under which ...