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

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41 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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2answers
45 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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0answers
29 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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0answers
18 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, ...
2
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27 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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1answer
64 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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0answers
13 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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0answers
37 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 ...
2
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2answers
23 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
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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0answers
26 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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24 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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2answers
93 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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2answers
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
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 ...
1
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1answer
37 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)= ...
1
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1answer
32 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 ...
3
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1answer
27 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 ...
0
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1answer
18 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
20 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 ...
2
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1answer
28 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
53 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
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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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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0answers
30 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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0answers
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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1answer
22 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
25 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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2answers
23 views

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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1answer
59 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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42 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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1answer
45 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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0answers
14 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 ...
2
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1answer
27 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\{ ...
3
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1answer
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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 ...
0
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1answer
19 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
45 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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1answer
22 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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3answers
125 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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1answer
48 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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1answer
16 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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1answer
88 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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1answer
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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16 views

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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1answer
43 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 ...
3
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1answer
74 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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2answers
64 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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1answer
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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2answers
25 views

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 ...
0
votes
1answer
58 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 ...