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 can I prove this operator is not continuos

Let $X$ the normed space of all polynomials on $J=[0,1]$ such that $||x||$=max$|x(t)|$ $t \in[0,1]$ and we have the following operator $Tx(t)=x'(t)$ prove this operator is not continuous
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10 views

Relation between max element norm and other norms of a matrix

I am looking for a relation between max element norm and other norms like $2$-norm or $\infty$-norm (in terms of bounds) so that I can reduce the following condition $$ ...
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1answer
47 views

How can I prove the following operator is continuous

Let we have the following operator $T$ From $C[a,b]$ to $C[a,b]$ by the formula $$Tx(t)=\int_{a}^{t}x(τ)dτ$$ How can I prove $T$ is continuous operator ?? The space $C[a,b]$ is squibbed with the ...
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23 views

Assuming $A$ is a nonexpansion in some norm, in what norm is $A^\top$ a nonexpansion.

Consider a matrix $A \in \mathbb{R}^{n \times n}$. Consider the vector norm $\| \cdot \|_\triangle = \| F \cdot \|_1$, where $F \in \mathbb{R}^{n \times m}$ and we have $m < n$ and $F$ has ...
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18 views

Is this inequality for finite sequences of numbers true?

Let $a_1,...,a_d$ be some positive reals with $\sum_{i=1}^d a_i \leq 1$. Let $x^k = (x^k_1,...,x^k_d)$ for $k=1,...,n$ be some vectors in $\mathbb{R}^d$. Is the following inequality true? And how is ...
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48 views

About a norm : $p(uv)=p(u)p(v)$ all the time? [closed]

Say, $p$ is a norm on a vector space. Then can we say that $$p(uv)=p(u)p(v)$$ all the time? Thanks.
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38 views

Find all $ p \ge 1 $ for which the Hölder norm $\|\cdot\|_p $ is generated by a scalar product.

Find all $ p \ge 1 $for which the Hölder norm $$ \|x\|_p := \left(\sum^{n}_{i=1} |x_i|^p\right)^{\frac{1}{p}} $$ is generated by a scalar product. We know that norm is generated by a scalar product ...
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18 views

Mahalanobis distance to the ellipsoid center

I am confused about the following description: If we parameterize the ellipsoid $E$ as: $E = \{x|\ ||Ax-b||_2 \leq 1\}$. $A \in S_{++}^n$ Then the Mahalanobis distance to the ellipsoid center is ...
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1answer
52 views

find linear functional norm

$C[-1,1] $ above $$ f(x)=\int_{-1}^{0}x(t)dt-\int_{0}^{1}x(t)dt$$ What is norm of the f linear fucntional? I tried to solve using definition of norm but I couldn't find result. Please can you give ...
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1answer
17 views

An infinite number of solutions available to the sparse representation problem

I would like to analyze the following problem (different cases leads to which solutions to the problem and such): $$||y-Dx||_2 \leq \epsilon$$ (an overcomplete dictionary matrix $D \in \mathbb{R}^{n ...
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53 views

Requirement on Norm for units in Cyclotomic Fields

Consider $\zeta_p$ a primitive $p^{th}$ root of unity. Prove that $\alpha \in \mathbb Z[\zeta_p]$ is a unit iff $N(\alpha)=\pm1$. I'm not even sure where to start with this. I know that ...
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1answer
45 views

Question about an inequality in a published paper which is yielded from an approximation

I am reading a published paper on K- SVD: An algorithm for designing overcomplete dictionaries for sparse representation In the introduction, it says: Recent years have witnessed a growing ...
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39 views

the spectral norm of the subtraction of two diagonal matrices

I saw this equation in some papers (e.g., http://cs-www.cs.yale.edu/homes/mmahoney/pubs/l2sample.pdf ), but I am wondering if it is wrong. The original equation is where $\|\|_2$ is the spectral ...
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1answer
26 views

Is it true that $|a_i| \leq |b_i|$ implies $ \|\Psi a \|_\infty \leq \|\Psi b \|_\infty$ for elementwise nonnegative $\Psi$s?

Denote by $\Psi \in \mathcal{P}$ the property that $\Psi$ has non-negative entries and independent columns. Does the following property hold for $a,b \in \mathbb{R}^n$. $(\forall i. |a_i| \leq ...
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52 views

Topology induced by norm

What is the meaning of topology induced by norm. To me topology is a collection of subsets satisfying certain rules. How can a norm induce a topology...? For example how can $\|\cdot\|_{2}$ induce a ...
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13 views

Bicomplex quadratic forms

The quadratic form associated with a real matrix $Q$ and a real vector $\vec{x}$ is $\langle Q \vec{x}, \vec{x} \rangle$. The quadratic form associated with a real matrix $Q$ and a complex vector ...
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1answer
88 views

Prove that the norm in an inner product space is $ \ge 0$

Macdonald Linear and Geometric Algebra defines an Inner Product Space in the following way (pg 57): "An inner product space is a vector space with a product called an inner product. The inner product ...
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27 views

Question about the norm $\|A \cdot \|_1$

$A \in \mathbb{R}^{n \times m}$, where $n > m$, has independent columns. Consider the vector norm $\| x \|_\Box = \| A x \|_1$. Consider two matrices $M_1$, $M_2$ which preserve this norm, i.e. ...
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29 views

Dimensionality of tangent vectors in R^2

I am puzzled with the following problem: given a tangent vector (a d/dx) in the Euclidean plane R^2 with "a" a dimensionless scalar, the dimensionality of this vector is, I suppose, 1/[lenght] and ...
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29 views

Why doesn't the $L_2$ norm differentiable at $x=0$?

Why doesn't the $L_2$ norm differentiable at $x=0$? Let's define $N(x)$ as the norm function. I know that for every $x\ne 0$: $$\frac{\partial N}{\partial x_i}(x) = \frac{x_i}{\|x\|}$$ What ...
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1answer
37 views

Determining Infinity Norm

I am trying to follow along with a practice problem in my book that asks me to determine the distance between two vectors $\textbf{u} = (-2, 2, 1)^{T}$ and $\textbf{v} = (1, 4, 1)^{T}$ using the ...
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1answer
44 views

Show that $F(x) = f(\|x\|)$ is differentiable on $\mathbb{R}^n$. [duplicate]

Let an even function $f:\mathbb{R}\to\mathbb{R}$ which is even and differentiable. We define $F:\mathbb{R}^n\to\mathbb{R}$ as $F(x) = f(\|x\|)$. Show that $F(x)$ is differentiable on ...
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1answer
52 views

Rank as norm on matrix

Could we consider matrix rank $r$ a norm? Is other norm similar to rank $r$ possible to associate with a finite matrix? (We denote trivial valuation $|\alpha|=1,\quad\forall\alpha\in\Bbb F^*$ where ...
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48 views

Bound on Symmetric Matrices

Let $A=(a_{ij})$ be a matrix with real entries, $1 \leq i, j, \leq n$. Let $A^{T}=(a_{ij}^{T})$ be the transposed matrix, that is $a_{ij}^{T}=a_{ji}$. Suppose that $a_{ij}=a_{ji}$, namely $A$ is a ...
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1answer
64 views

Hessian of Frobenius norm

I want to find the Hessian of the following function, $F(\mathbf{X}) = \frac{1}{2}||\mathbf{Y} - \mathbf{AX}||_F^2$. My try: Using trace formula for Frobenius norm, $F(\mathbf{X})$ can be written as, ...
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24 views

Why is $\ell_{2,1}$ mixed norm non-smooth?

I'm reading about optimization problems involving mixed norms. In particular I'm getting acquainted with the $\ell_{2,1}$ norm. For a matrix $\mathbf{X}$, the $\ell_{\alpha,\beta}$ norm, ...
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34 views

Equivalence of norms between two Banach spaces

I just learnt open mapping theorem. And I met a statement online asserting that If $X$,$Y$ are Banach space, and $T:X\to Y$ be a continuous bijection, then norms for $X,Y$ are equivalent. Can we ...
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71 views

Norm of the product of an isometry and a bounded operator

Let $A$ be a bounded operator and $V$ a linear isometry, both defined on a complex Hilbert space $H$ (infinite dimensional). I could easily prove that $\|VA\|=\|A\|$. But, I just couldn't prove that ...
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16 views

Pullback of the norm on the holomorph by the Riesz functional calculus

Conway states that the holomorph $H(a)$ of an element $a$ of a Banach algebra is not a Banach algebra. Let $||f||=||f(a)||$ for any $f\in H(a)$. We need to see that this "norm" is separates the ...
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43 views

Matrix Norm Inequalities

I need to prove $\frac{1}{\sqrt{n}}||A||_2 \leq ||A||_1 \leq \sqrt{m}||A||_2$, where $A$ is a $m\times n$ matrix. I think it has something to do with higher order norms being less than or equal to ...
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27 views

Squaring Norms solved by Algebra

I found the following in a paper and am not sure how it is correct: $\Vert a - b \Vert^2$ was expanded to: $\Vert a \Vert^2 - 2a^Tb + \Vert b \Vert^2$ The paper was on gps location algorithms so ...
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1answer
29 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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27 views

why and when t0 use norm instead of abs and vice versa

What is the difference between the norm and abs of an expression.. as far i understand does ||a - z|| mean norm and |a-z| abs , but what is the difference?
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33 views

What is the default norm on a matrix?

In this article, in section 3.5, a step size $\delta^i$ is calculated using, amongst other things, what I assume is the norm on a matrix $g^i$ ($||g^i||$). However, no mention is made of how this norm ...
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119 views

What are the best books for studying functional analysis in the world

I want to ask you maybe strange question but I really need answer What are the best books for studying functional analysis After Afew week I start study in master so I want references
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26 views

Norm controls the components of a matrix- proof?

For certain norms (such as the Frobenius norm) it is clear that the norm provides component-wise control - each component is at most as large as the norm in magnitude. How do we establish this for ...
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1answer
35 views

Apply Cauchy-Schwarz 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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1answer
39 views

I can't understand one step of this matrix norm proof

I'm reading a multivariable calculus textbook for college, and before a Taylor series proof is given, a lemma is provided with its demonstration. The lemma says: Given a matrix $M(X)= ...
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1answer
40 views

Bounding a Symmetric Matrix

Consider the following $n \times n$ matrix $A$, which has 1's on the superdiagonal and subdiagonal and 0's elsewhere, i.e. $$\begin{pmatrix} 0 & 1 & 0 & \cdots & \cdots & \cdots ...
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1answer
39 views

Definition of the norm of a bounded linear operator.

I have a somewhat basic but confusing question regarding the definition of the norm for bounded linear operator. Suppose $f$ is a bounded linear operator, that is, there exists $M>0$ such that ...
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1answer
24 views

Definition of an induced matrix norm.

Could someone explain the second equality in the definition of a induced matrix norm to me? Let $\| \cdot \|$ be a norm for $\cdot \in \mathbb{R}^n$, then the induced matrixnorm for $A\in ...
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1answer
21 views

Let $u,v \in \mathbb{R}^n, A = u\cdot v^T$ then prove $\|A\|_2 = \|u\|_2 \; \|v\|_2$

Let $u,v \in \mathbb{R}^n, A = u\cdot v^T$ then prove $\|A\|_2 = \|u\|_2 \; \|v\|_2$ How can I do this? I'm kinda stuck with this 2-norm of the matrix. If it would be $\|A\|_{\text{frob}}$ then ...
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1answer
29 views

For invertible $A$ show that $\lbrace y \in \mathbb{R}^n : \| x-y \|_A < r \rbrace= \lbrace x + A^{-1} y: y \in B_r(0) \rbrace$

I am struggling with the following Problem: Let $| \cdot|$ be the Euclidian Norm on $\mathbb{R}^n$. Let $A$ be an invertible $n \times n$ Matrix. Define $\|x\|_A = |Ax|$ for $x \in \mathbb{R}^n$ ...
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1answer
164 views

Show that the dual norm of spectral norm is Nuclear norm.

Could someone help to understand that the dual norm of spectral norm is Nuclear Norm ? We can focus on the real field. Given a matrix $X \in \mathbb{R}^{mn},$ then the spectral norm is defined by: ...
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1answer
27 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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1answer
31 views

Does $||x|| < c$ imply $|x_i| < c$?

Given an $x=(x_1,\dotsc,x_n) \in \mathbb{R}^n$ for some $n \in \mathbb{N}$ and an arbitrary norm $||\cdot||$ Does $||x|| < c$ imply $|x_i| < c$ for some $c > 0$?
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44 views

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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40 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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1answer
25 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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1answer
30 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 $l(v)$), ...