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

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59 views

Norms of eigenvalues bigger than 1 implies $|Ax|>x$ for all nonzero $x$?

If all the eigenvalues of $A$ (an n by n real matrix) have norms bigger than 1, is it true that $|Ax|>|x|$ for all nonzero $x\in\mathbb{R}^n$? (This is clearly true if $x$ is an eigenvector ...
7
votes
1answer
244 views

Does this cross-product norm inequality hold?

Let $\times$ denote the cross-product. $\;$ Is it the case that For all unit vectors $\:\mathbf{x}\hspace{.01 in},\hspace{-0.03 in}\mathbf{y}\hspace{-0.03 in},\hspace{-0.02 in}\mathbf{z}\:$ in ...
-1
votes
1answer
571 views

Sobolev space - norm $H^1$ and $H^1_0$

When we defined on $H^1_0$ the norm $$||v||_{H^1_0}=||v||_{L^2}+||\nabla v||_{L^2}$$ can we tell that $$||u||_{H^1_0} = ||u||_{H^1}?$$ Thank's
0
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3answers
87 views

Proving nonsingularity of a sum of matrices

I'm trying to solve this study question but I'm not sure how to proceed. The question is as follows. If \begin{equation}\frac{||B||_2}{||A||_2}<\frac{1}{\kappa_2(A)}\end{equation} with ...
2
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4answers
140 views

Prove that $||x|-|y|| \leq |x-y|$ [duplicate]

$||x|-|y|| \leq |x-y|$ when $(x,y \in R^k)$ In Principles of MA(Rudin), the author said one sees easily that $||x|-|y|| \leq |x-y|$ when $(x,y \in R^k)$ (p.88, Rudin) from the triangle ...
3
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3answers
103 views

Prove an inequality.

Prove that $$\displaystyle{(|x_1+y_1|^p + |x_2+y_2|^p +\dots +|x_n+y_n|^p)^{\frac{1}{p}}\leq (|x_1|^p + |x_2|^p +\dots +|x_n|^p)^{\frac{1}{p}}+(|y_1|^p + |y_2|^p +\dots +|y_n|^p)^{\frac{1}{p}}}$$ for ...
0
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1answer
77 views

What is a decreasing scale of Banach spaces?

I am a having a hard time understanding a part of an article I am reading. The screen-cap is below. Basically, it's the line labeled (6) that I do not understand. I am not familiar with the circular ...
1
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1answer
81 views

What is an operator norm?

I came across this symbol where they are mentioning operator norms. I am not sure how it is different from the Frobenius norm. It was something like this: $|||\Omega-\hat{\Omega} |||_2$ where ...
0
votes
1answer
68 views

Extrema of a vector norm under two inner-product constraints.

If $\langle\vec{A},\vec{V}\rangle=1\; ,\; \langle\vec{B},\vec{V}\rangle=c$, then: \begin{align} max\left \| \vec{V} \right \|_{1}=?\;\;\;min\left \| \vec{V} \right \|_{1}=? \end{align} Consider the ...
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0answers
122 views

Convergence of the Jacobi iteration method

I think I am not quite understanding the Jacobi Method or some related concept for indirectly solving linear systems of equations of the form $Ax=b$. We need the norm $||I-Q^{-1}A||_\infty < 1$ and ...
2
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1answer
84 views

How to force unitary Euclidean norm in a complex matrix by multiplication with a diagonal matrix

I need to solve the following problem: Suppose a non-sparse, non-singular complex matrix $\mathbf{P}$. If I want to force all rows in $\mathbf{P}$ to present unitary Euclidean norms by multiplying ...
0
votes
1answer
304 views

Norm of a vector-valued function?

When studying commutator estimates, I have encountered the following problem. Consider $f\in C^1(\mathbb{R}^d,\mathbb{R})$ with $\nabla f\in L^p$. So $\nabla f(x)\in\mathbb{R}^d$. My question is ...
2
votes
2answers
87 views

What is the definition of the norm

Let $x$ and $y$ be in $ \mathbb{R}^{n}$. I know from the definition of norm that $\|x\|=\sqrt{\sum_{1}^{n}x_{i}^{2}}$. Can anyone tell me what will be the norm of $\|x-y\|$? Is it ...
2
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2answers
71 views

Generalizing norms for modules

An normed space is defined as a vector space V plus a norm operation over $V$. Is it meaningful to generalize this notion to modules, where one is dealing with rings instead of fields? What I'm ...
1
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1answer
96 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 ...
3
votes
1answer
102 views

Show if $\|\cdot\|$ is a norm then $\|f(\cdot)\|$ is a norm where $f$ is linear and invertible

I want to show that if $\|\cdot\|$ is a norm then $\|f(\cdot)\|$ is a norm where $f$ is a linear and invertible function. First I need to show if $x\neq0$ then $\|f(x)\|>0$. Since $f$ is ...
0
votes
1answer
100 views

Showing a norm preserving isomorphism of vector spaces

Lets define $l^1$ as the complex vector space of all absolutely summable sequences of complex numbers and and $x_0$ consists of all the sequences in $l^\infty$ (all bounded sequences) that eventually ...
0
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1answer
174 views

Showing triangle inequality for a norm

I want to determine whether the following is a norm or not: \begin{equation} ...
1
vote
1answer
165 views

Proving the triangle inequality for the L-2 norm $||x||_2 = \sqrt{x_1^2+x_2^2\ldots+x_n^2}$

I want to prove the triangle inequality for the l2-norm $||x||_2$: $$||x||_2 = \sqrt{x_1^2+x_2^2+\ldots+x_n^2}$$ $$\begin{align} \sqrt {\sum\limits_{i = 1}^n {{{\left( {{x_i} + {y_i}} \right)}^2}} ...
1
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1answer
212 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 ...
1
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2answers
332 views

Property of Subordinate Matrix Norm: $\|AB\| \leq \|A\|\|B\|$

I do not understand why the following property for Matrix subordinate norms holds: \begin{equation} \|AB\| \leq \|A\|\|B\| \end{equation} Please explain clearly as I know that it should be shown by ...
0
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1answer
680 views

How to describe minimization of L1 norm error using linear programming?

Given a set of $n$ pair points $(x_1, y_1), ..., (x_n, y_n)$ in the plane, I need to find a line $ax + by = c$ that fits the points of the L1 norm error points as closely as possible. I need a linear ...
5
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2answers
265 views

Why is the 2 norm “special”? [duplicate]

Out of all the vector norms, the $2$ norm, or the Euclidean norm, seems to be "special". Primarily, I say this because we use the 2 norm as a means of determining the distance from one point to ...
2
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1answer
222 views

Property for Norms of Matrices

I am having trouble with the following problem: Show that the vector norm $||x||_1$ gives the subordinate matrix norm: \begin{equation} ||A||_1 = \max_{1\leq j\leq n}\sum_{i=1}^n|a_{ij}| ...
0
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2answers
169 views

Variation of reverse triangle inequality

I know from the reverse triangle inequality that for $x,y \in \mathbb{R}^n$ the following holds: $ \vert x \vert - \vert y \vert \leq \vert x -y \vert $ but does also this one hold? $ \vert x ...
3
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1answer
63 views

Showing that the zero vector has norm zero

I need to show that this is a property of a norm. I know this is supposed to be straightforward but I am somehow not seeing it. The property is $$\lVert 0\rVert = 0$$ I was trying to use the fact ...
2
votes
1answer
90 views

Derive $\frac1n \|x\|_p^p \leq \|x\| \leq n^{p/2}\|x\|_p^p$ from Holder's inequality?

Given a vector $x = (x_1, \dotsc, x_n)\in \mathbb{C}^n$, I wanted to compare $|x_1|^p + \dotsb + |x_n|^p$ to $\|x\|^p$. I discovered that if $m=\max_i|x_i|$, we have $$m^p \leq \|x\|^p \leq ...
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0answers
70 views

Distance of a point from a subspace vs. diameter

Let X = $(\Bbb R^N, \|\cdot\|)$ be a Banach space. Let $x_0 \in S^{N-1} = \{x \in \Bbb R^N : \sqrt{x_1^2+...+x_n^2}=1\}$. Denote $B^N_2 = \{x \in \Bbb R^N : \sqrt{x_1^2+...+x_N^2} \le 1 \}$. Define ...
0
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1answer
54 views

Matrix expansion does not decrease norms

Given a block matrix $A = \left[ {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}}\\ {{A_{12}}}&{{A_{22}}} \end{array}} \right]$, where $A \in {R^{N \times N}}$, it is true that the euclidean ...
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0answers
30 views

Norms for minimizing arbitrary error distribution

Assume we are doing regression on the function $f(x)$ with error term $e(x)$ with distribution $g(e; \theta)$: $ y = f(x) + e(x), \; e(x) \sim g(e; \theta) $ Let's say we know the analytic form of ...
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1answer
38 views

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) = ...
2
votes
1answer
2k views

How to prove triangle inequality for $p$-norm?

Well, I've been studying metric spaces and to make the cartesian product of metric spaces a metric space I've heard of the $p$-norm defined in $\mathbb{R}^n$. So if $\mathcal{M}=\{M_i : i\in I_n\}$ is ...
1
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2answers
185 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. ...
2
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1answer
135 views

Norm of Hilbert's operator $H f(x)= \int_0^{\infty} \frac{f(y)}{x+y}\, dy$ [duplicate]

Hilbert's operator $$H f(x)= \int_0^{\infty} \frac{f(y)}{x+y}\, dy \quad\text{ for all } f \in {L}^2(0,+\infty) \text{ and } x \in(0,+\infty),$$ is regular integral operator on $L^2(0,+\infty)$ ...
1
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1answer
212 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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2answers
330 views

Proving that $\|A\|_{\infty}$ the largest row sum of absolute value of matrix $A$

I am studying matrix norms. I have read that $\|A\|_{\infty}$ is the largest row sum of absolute value and $\|A\|_{1}$ is the highest column sum of absolute values of the matrix $A$. However, I am ...
1
vote
3answers
114 views

Why is $\frac{x}{\| x \|}$ a unit vector? [duplicate]

Let $x$ be a vector in $\mathbb{R}^n$. Why is $\frac{x}{\| x \|}$ a unit vector, for $x\neq 0$? If I try to simplify it, I get the following: $\frac{x}{\sqrt{x \cdot x}}$, and I'm not sure how to ...
0
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2answers
79 views

Mysterious Matrix Norm

Given a matrix $M$, does anyone know the name and the definition of the following norm? $$ \|M\|_* $$ Thanks in advance, Francesco.
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2answers
133 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 ...
2
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1answer
76 views

Calculating the norm of $S\otimes T$ for bounded linear $S,T$.

Let $X,Y,W,Z$ be Banach Spaces. Let $X\otimes_{\pi}Y$ denote the tensor product endowed with the projective norm. If I have $S\in B(X,Z)$, $T\in B(Y,W)$, it is straightforward to show that $S\otimes ...
0
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1answer
52 views

SVM - Min square norm

All Support Vector Machine litterature mentions that optimal hyperplane is found as: max 1/∥x∥ (st. constraints) which translates directly to: min ∥x∥ or equivalently min $ ∥x∥^2 $. Here ...
2
votes
1answer
449 views

Why do we need semi-norms on Sobolev-spaces?

I have been studying Sobolev spaces and easy PDEs on those spaces for a while now and keep wondering about the norms on these spaces. We obviously have the usual norm $\|\cdot\|_{W^{k,p}}$, but some ...
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2answers
549 views

show operator norm submultiplicative

We had in our lecture on numerical analysis the following: Let $\mathrm{Lin}(X,Y)$ be the set of all linear maps $X\rightarrow Y$. Let $A\in\mathrm{Lin}(\mathbb R^l,\mathbb R^n)$ and ...
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0answers
26 views

discrete version of $L_p$

Why is $(\sum_{i=1}^{N-1}e^{-px_i/\epsilon}\bar{h}_i)^{1/p}=O(N^{-1/p})$,as $N$ approaches 0, where $\bar{h}_i=(h_{i+1}+h_i)/2$, $h_i=x_i-x_{i-1}$? The integral version is much easier to calculate.
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42 views

Natural invariant norm on the space of polynomials

What is the " Natural" invariant norm on the space of polynomials in a complex variable $z$? And can anyone give me an idea as to how it is deduced?
3
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2answers
95 views

When does $\|x+y\|=\|x\|+\|y\|?$

Let $(V_\mathbb R,\langle,\rangle)$ be an inner product space. I'm trying to see for $x,y\in V$ when does $\|x+y\|=\|x\|+\|y\|?$ Let $\|x+y\|=\|x\|+\|y\|$ Squaring both sides, $\langle ...
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1answer
120 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 ...
3
votes
1answer
132 views

Norm of a linear transformation

Let $T:\mathbb R^2\to \mathbb R^2$ be given by the matrix $\begin{pmatrix}a&b\\ c& d\end{pmatrix}$. Let $u:=a^2+b^2+c^2+d^2+2(ad-bc)$ and $v:=a^2+b^2+c^2+d^2-2(ad-bc)$. I need to show that ...
2
votes
0answers
102 views

Operators from $L^{\infty}$ to $L^{\infty}$, below bound of the norm

If $T(f)(x)=\int K(x,y)f(y)dy$, where $K(x,y)$ is locally integrable, is bounded on $L^{\infty}$, how can we show that $\|\int|K(\cdot,y)|dy\|_{L^{\infty}}\le \|T\|_{L^{\infty}\rightarrow ...
1
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1answer
212 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 ...