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

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does the max function holds the triangle inequality?

I need to prove if the following is a norm: $$||f||:=\max_{-1<x<0}|f| + \max_{0<x<1}|f|$$ when $f$ is a continuous on $[-1,1]$. The only problem I have is with showing it holds the ...
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57 views

Space on convergent sequences - is it an inner product space

I'm trying to prove some properties of sequence spaces. I already know that the space $l^{\infty}$ of all bounded sequences isn't an inner product space, isn't separable but it is complete with $sup$ ...
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159 views

Bound on the norm of a matrix exponential in Jordan Form

I'm looking to prove the following lemma: Let $A$ be a matrix in $\mathbb{R}^{n\times n}$. Then for any $\lambda^* > \max_{\lambda} \; \mathrm{Re} \; (\lambda)$ such that $ \lambda \in\sigma (A)$, ...
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63 views

Norm of convolution of $n$ Gaussians

If $$f(x)=e^{-(\pi x)^2}$$ and $$\psi_n(x)=(f* f*\dots*f)(x)$$ ($n$ times convolution). Show that $$\lVert \psi_n(x)\rVert = 1$$ (norm in $L^1(\mathbb{R})$). I've tried using the Fourier ...
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70 views

How an $\ell_1$ Inequality Implies Equality

Suppose that for scalar $\epsilon$ we know that $\vert \epsilon \vert$ is small enough such that the sign pattern on $\mathbf{x}\in\mathbb{R}^n$ is equal to that on $\mathbf{x} + \epsilon \mathbf{h}$, ...
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54 views

How to show Zygmund space is Hölder space?

The motivation of this question is to show that Zygmund space is Hölder space, in certain cases. For simplicity, take $s\in (0,1)$, I want to show $$\|f\| = \|f\|_\infty + \sup_{x,y\in ...
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148 views

Convexity of the squared Frobenius norm of a matrix

I was reading this paper where the define an optimization problem as where K and L are kernel matrices and $\pi$ is the permutation matrix. They have explained that the function is convex because ...
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153 views

norms and sparsity

Could anyone please elaborate on why $L^2$ norm moves toward the outliers compared to $L^1$ norm. I mean, what property/quantity in the mathematical expression of the norms makes it perform such way. ...
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167 views

Norm of integral operator

Consider the operator $T(f(t)) = \int_0^t f(s)ds$, where $t \in [0,1]$, and $f(t) \in C[0,1]$. To prove $$\|T^n\| = \frac{1}{n!}$$ Thanks for suggestions.
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77 views

What is magnitude of sum of two vector?

I know that magnitude of $\bf X$ is defined as: $$||\bf X||=\sqrt {(\bf {X\cdot X})}$$ Now if I define $\bf X$ as the sum of two vector like this $\bf X=\bf X_1+\bf X_2$ then what will be the ...
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97 views

Is the bound between the matrix 2-norm and the max-norm tight?

It is well known that $\|A\|_2\leq\sqrt{mn}\|A\|_{\max}$ for an $m\times n$ matrix. Is this bound tight? i.e which matrix $B$ satisfies $\|B\|_2=\sqrt{mn}\|B\|_{\max}$ (note the equality)? And is ...
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167 views

Bounded operator inverse, norm and spectrum

I need help with an operator, I am not very good at functional analysis and need to find some properties of following operator: $X=C[(0,1)], A \in B(X); A[f(t)]=f(t^2)$ 1. I need to show that an ...
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39 views

Homogenous Systems Question

$A, B \in \mathbb{C}^{n\times n}$ If $I - B$ is singular, then $\exists x$ from $\mathbb{C}^n$ so that $(I - B)x = 0$. How do I show that $|| B || \geq 1$ and if $|| A || \leq 1$ then $I-A$ isn't ...
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48 views

Lipschitz condition normed vector space

Am I right that $g: C^{1}[0,2],||*||_{C^1[0,2]} \rightarrow \mathbb{R}$ with $g(f)=f'(1)$ satisfies a Lipschitz condition? Cause $|f'(1)-h'(1)|=|g(f)-g(h)|\le \text{max} |f-h|+ \text{max} |f'-h'|$, ...
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215 views

What is the role of supremum in operator norm

An operator norm is defined as $\|A\|_S=\sup\{\|Av\|:v\in \Bbb R^n, \|v\|=1\}$. Where $\|\cdot\|$ is some norm on $\Bbb R^n$ and $A\in M_n(\Bbb F)$, space of square matrices of dimension $n$ over ...
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78 views

Study the equivalence of these norms

I have two Hilbert spaces $H_1$ and $H_2$ and I consider a set of functions $f$ which decompose as $f=g+h$ with $g\in H_1$ and $h\in H_2$. I know that this decomposition is unique. So I define the ...
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98 views

On norm selection for the solution of an overdetermined linear system

I am considering the following linear system: $Ax = b$ Where: $A$ is $9000 \times 139$ $x$ is $139 \times 1$ and sparse $b$ is $9000 \times 1$ Most of the resources I have found online point to ...
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44 views

Order of infinite dimension norms

I know that $$\|{f}\|_{L^1(0,L)}\leq\|{f}\|_{L^2(0,L)}\leq\|{f}\|_{\mathscr{C}^1(0,L)}\leq\|{f}\|_{\mathscr{C}^2(0,L)}\leq\|{f}\|_{\mathscr{C}^{\infty}(0,L)}$$ But I don't know where to put in this ...
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46 views

Determining a norm from a quadratic form

If $B$ is a quadratic form over some space $V$, what is the norm determined by $B$? Is this the inner product $\langle Bu,Bv\rangle$? If not, and it is not possible to determine a norm from knowing ...
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81 views

How to show for a PSD matrix $A$ that $\left \| \left ( A+I \right )^{-1} \right \| \leq \frac{1}{1+\sigma _{\min}\left ( A \right )}$?

If $A \in \mathbb{C}^{n \times n}$ is positive semidefinite, show that $\left \| \left ( A+I \right )^{-1} \right \| \leq \frac{1}{1+\sigma _{\min}\left ( A \right )}$, where $\sigma _{\min}\left ( A ...
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109 views

Does π depends on the norm? [duplicate]

If we take the definition of π in the form: π is the ratio of a circle's circumference to its diameter. There implicitly assumed that the norm is Euclidian: \begin{equation} ...
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62 views

A bound on the norm of the sum of two index-disjoint matrices

Given two matrices, it is well known that $\parallel A+B \parallel _2 \leq \parallel A \parallel _2+\parallel B \parallel_2$. Now, suppose that the nonzero indices are disjoint (i.e., $A$ is nonzero ...
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271 views

Change in singular values of matrix after left-multiply with a diagonal matrix

Say that we have an SVD for a matrix $X = U \Sigma V^T$, giving trace norm $||X||_{tr} = ||\Sigma||_{tr} = \sum \Sigma_{ii}$. I am wondering what happens to the SVD and/or trace norm if we left ...
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203 views

Norms of idempotent matrices

So if I have a random idempotent matrix (a matrix that when squared equals itself) how do I go about calculating its 2-norm? I know that a idempotent matrix has eigenvalues of only 0 or 1 and I know ...
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114 views

Matrix Trace representation?

For a real, symmetric matrix $A$ and a real, rectangular matrix $X$, am looking for a matrix trace based representation of this simple linear algebraic expression $\sum_{i} A_{ii} ...
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65 views

Norm of Polynom

I'm trying to understand the following equation: $$\langle p,q\rangle_p=\int_{-1}^1p(x)q(x)dx\\ \text{Basis: }\{p_1,p_2\}\\ p_1:=2x,\quad p_2:=x-1$$ Norm: $$p_1:q_1=\frac{p_1}{\|p_1\|_p}$$ ...
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162 views

Equivalence of Norms Defined on a Cartesian Product

While studying some notes on normed vector spaces, I have come upon the proof that addition $+:V \times V\to V$ of vectors in a normed vector space $V$ is a continuous operation. The proof of this ...
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99 views

norm of a matrix ( which norm have to use ?)

I need to find the norm of the matrix $$ A=\left( \begin{array}{cc} e^{-x} \cos( \sin x) & e^{-x} \sin ( \sin x) \\ -e^{-x} \sin ( \sin x) & e^{-x} \cos (\sin x) \end{array} \right) $$ Here ...
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99 views

Small question regarding norms and Holder conjugates.

I'm trying show that if $p,q$ are Holder Conjugates then: $$\forall\, a\in\mathbb{R}^{n}:\,\Vert a\Vert_{q}=\max_{x\in\mathbb{R}^{n},\,\Vert x\Vert_{p}=1}\left<a,x\right>$$ Where ...
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why is frobenius norm of a matrix greater than or equal to the 2 norm?

How can you prove that: $$\|A\|_2 \le \|A\|_F$$ I cannot use: $$\|A\|_2^2 = \lambda_{max}(A^TA)$$ It makes sense that the 2-norm would be less than or equal to the frobenius norm but I dont know how ...
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prove matrix norm equivalence

Given $A \in R^{m\times n}$, I need to prove: $$||A||_2 \le \sqrt {m}||A||_\infty$$ I have tried a number of things and I just cant seem to get it to work. Also, I need to prove: $$||A||_2 \le ...
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49 views

Does switching between different $L_p$ norms preserve order?

Suppose you have a list of $n$ dimensional vectors. One can order them by using an $L_p$ norm to do comparisons between vectors. The general questions is, will the order be different depending on the ...
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85 views

Estimating the integral in norm.

I want to estimate the integral $$\int k(x,y)f(y)dy$$assuming the fact that $k(x,y), f(y)$ are in $L^p, L^q$ respectively. But I want to bound the the whole integral in $L^r$, $r\in [1,\infty]$. I ...
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189 views

operator norm and spectral radius

is it true that the operator norm of a matrix $A$ is smaller than 1 if its spectral radius $\rho(A)$ is smaller than 1? many thanks for any help, it is much appreciated!
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How to calculate norm of operator in Hilbert space

LEt $H_1,H_2$ are two Hilbert spaces. $\{e_1,\ldots,e_n\}\subseteq H_1$ and $\{f_1,\ldots,f_n\}\subseteq H_2$ two orthonormal systems. $\lambda_1,\ldots,\lambda_n\in\mathbb K$. Let $$ U:H_1\to ...
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67 views

Could Ky-Fan Norms Improve the Bound on the Max Norm of $A$

Let $A$ be real symmetric and $D$ shall contain the eigenvalues of $A$. I've learned that $\|A\|_{\text{max}}< \|D\|_{\text{max}}$, where $\|A\|_{\text{max}}$ means the Max norm. I want to get a ...
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68 views

what does it mean to say a space is norm separable?

I came across in my textbook the term: norm separable. I looked in the textbook and online and could not find a definition.
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33 views

How are $\|A\|_{\text{max}}$ and $\|D\|_{\text{max}}$ related?

Max norm The max norm is the elementwise norm with $p = \infty$: $$ \|A\|_{\text{max}} = \max \{|a_{ij}|\}. $$ This norm is not sub-multiplicative. Let $A$ be real symmetric and $D$ ...
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123 views

essential supremum of a matrix multiplication operator

Suppose we have the space $L^p(R,R^n)$ where $1 \leq p < \infty$ (i.e the space of functions that take values in $R^n$ and are $L^p$ integrable) and suppose $T_m: L^p(R,R^n) \to L^p(R,R^n) $ is a ...
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777 views

Gradient of squared Frobenius norm

I'd like to find the gradient of $\frac{1}{2} ||X A^T||_F^2$ with respect to $X_{ij}$. Going by the chain rule in the Matrix Cookbook (eqn 126), it's something like $\partial \left[\frac{1}{2} ||X ...
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245 views

The openness of the set of positive definite square matrices

Let $\mathbb{R}^{n\times n}$ be the vector space of square matrices with real entries. For each $A\in \mathbb{R}^{n\times n}$ we consider the norms given by: $$ \displaystyle\|A\|_1=\max_{1\leq j\leq ...
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190 views

Equivalence of Schatten and spectral norms

I'd like some help showing the equivalence of these two norms when $p = \log n$. Recall the $p$-th Schatten norm of a linear operator $A$ acting on $\mathbb{R}^{n}$. In the particular case of $p = ...
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239 views

equivalence of two definitions of norm equivalence: “$|\cdot|_1=|\cdot|_2^\alpha$” vs. “being a Cauchy sequence is the same for both norms”

Let $|\cdot|_1$ and $|\cdot|_2$ be two norms on a field $\mathbb F$. We call the two norms equivalent if every Cauchy-sequence with respect to $|\cdot|_1$ is also a Cauchy-sequence with respect to ...
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172 views

Representing with Hilbert Schmidt Norm

Am trying to see, if the following Trace function can be expressed using a Hilbert Schmidt Norm: $\operatorname{Tr}(X^TAX)$. Here, $X$ is a matrix whose entries take values that are finite and reals ...
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299 views

Time complexity of norm function in Matlab

What is the time complexity of the "norm" function in matlab?
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170 views

p-norm of series / Mathematica

When computing the $p$-norm of the series $x_n = \mathrm e^{\mathrm i n}$ for $n \in \mathbb{Z}$ (not $\mathbb{N}$), the sum should not converge: $$\|x\|_p^p = \sum_{n \in \mathbb{Z}} (\mathrm ...
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89 views

What is the proper term for the entity that relates a vector space and a set?

One way to generate a metric for a set $S$ (a distance function between elements $a,b$ of the set $S$) would be by associating it with a vector space $V$ (the vectors that connect the elements $a,b$) ...
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60 views

Equivalence of a Vector Norm being Absolute

I'm trying to show that a vector norm $\|\cdot\|$ being absolute ($\|x\| = \|\;|x|\;\|)$ is equivalent to showing that $\|x'\| = \|[\alpha_1x_1\ldots\alpha_nx_n]^T\| = \|x\|$ for all ...
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59 views

Operator norm of the inverse

If I made no mistake, one can calculate the operator norm of the inverse of any given (invertible) operator $A: V\rightarrow V$ via: \begin{align}\|A^{-1}\| & = ...
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Immediate consequence of the definition of Operator Norm. Explain

||Av|| $\leq$ ||A||$_{op}$||v|| for every v $\in$ V I was wondering why this is true. Wikipedia says it's an immediate consequence of the definition but I just do not follow. I am using the ...