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

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Norm of an operator and eigenvalues

I have $K\colon L^2(0,T) \to L^2(0,T)$ a Hilbert-Schmidt integral operator (and so $K$ is linear, bounded, compact and self-adjoint) and I have obtained its eigenvalues and eigenvectors. From them, I ...
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74 views

$L^1$ norm of convolution

Let $f_{\lambda} = \frac{\lambda}{2}e^{-\lambda |x|}$. Prove that $||f_{\lambda} \ast g - g || \to 0$, when $\lambda \to \infty$, where $g \in L^1$
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129 views

Inequality of Frobenius norm for skew matrices

Let $A$ be a complex skew-symmetric $n \times n$ matrix, that is, $A^T = -A$. Denote by $\|\cdot\|_F$ the Frobenius norm, that is, $\|B\|_F^2 = \text{trace}(B^*B)$. I would like to prove that $$ ...
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70 views

Hilbert-Schmidt norm/smooth manifolds

Given two riemannian manifolds $M$ and $N$ and a smooth map $f$ : $M$ $\rightarrow$ $N$, we define the energy density of $f$ as the smooth function $e(f)$ : $M$ $\rightarrow$ $\mathbb{R}$ given by ...
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200 views

Norm- Orthogonal Projection $A^{t}A=I_{n}$

Let $A_{m\times n}$ be a matrix such that $m\geq n$ and $A^{t}A=I_{n}$. It is given that the columns of $A$ are orthonormal. I need to show that the 2-norm of each row of $A$ is $\leq 1$. I have no ...
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continuity of norms with respect to $p$

Let $f\in L^{\infty}(\Omega,\Sigma,\mu)\cap L^{1}(\Omega,\Sigma,\mu)$. Then $w(p)=||f||_p$ is continuous function of $p$ for any $p\in [1,\infty)$. How to prove this? I have obtained the proof that ...
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evaluating norm of sum of roots of unity

let $l_1,...,l_n$ be roots of unity. I want to prove that the norm(the product of all conjugates)of $a=l_1+...+l_n$ is not greater than $n$, not smaller than $-n$. how can I do to prove this?
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227 views

Finding matrix norm equivalence constants

I've been given the following: "Find the best positive constants $\alpha$ and $\beta$ such that $\alpha\left\|A\right\|_2\leq\left\|A\right\|_1\leq\beta\left\|A\right\|_2$ for all ...
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188 views

Closed-form expressions for dual norms of real normed vector spaces

Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by $\|v\|$. Then say that $V^*$ is the dual space of linear functionals on $V$. The "dual ...
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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 ...
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332 views

Finding operator norm

I have to solve the following problem: Find a norm of operator $$A:L^2[-\pi,\pi]\rightarrow L^2[-\pi,\pi]$$ given with $$Af(x)=\int_{-\pi}^{\pi} \cos^2{\left(\frac{x-t}{2}\right)}f(t) \,dt.$$ I ...
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48 views

bound on $l_2$ error in approximating a vector with its $t$-sparse representation

How do I prove that for any vector $y\in \mathbb{R}^n$, and any positive integer $t$, \begin{equation} ||y-y_t||_2\:\leq\: \frac{1}{2\sqrt{t}}||y||_1 \end{equation} where $y_t\in\mathbb{R}^n$ is the ...
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How to show that $\left \| AB \right \|_F \leq \min (\left \| A \right \|_F \left \| B \right \|_2 , \left \| A \right \|_2 \left \| B \right \|_F)$?

For any matrices $A \in \mathbb{C}^{m \times k}$ and $B \in \mathbb{C}^{k \times n}$, show that $\left \| AB \right \|_F \leq \min (\left \| A \right \|_F \left \| B \right \|_2 , \left \| A \right ...
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Absolute norms and 1-unconditional sums

Absolute norm Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that $$ \|(x,y)\|_N=N((\|x\|, ...
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173 views

Norm of operator $g\mapsto \int fg$

Let $T_f:(C([a,b],\mathbb C), \lVert \cdot \rVert_1) \to \mathbb C$ with $g\mapsto \int_a^b f(x)g(x) dx$ for any given $f\in C([a,b],\mathbb C)$. I have to find the norm of $T_f$. I started with: ...
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276 views

Norm with special conditions

Let $N$ be a norm on $\mathbb R^2$, such that $N ( \mathbb Z^2) \subset \mathbb N $, where $\mathbb Z^2 =\{ (a,b)\mid a\mbox{ and }b \mbox{ are integers}\}$. Help me to prove that for $u$, $v$ fixed ...
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80 views

Inequivalent norms (given by different inner products) on infinite dimensional Hilbert space.

I have this question in reviewing for my exam. Let $H$ be an infinite dimensional Hilbert space. Write down an inner product on $H$ that gives a norm inequivalent with the original norm. Is $H$ ...
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75 views

Do these limits commute?

Given a sequence of functions $f_{n,m}:\mathbb{R}^{n} \to \mathbb{R}$, suppose that $$\displaystyle lim_{m} f_{n,m}(x)$$ exists almost everywhere (for any fixed n) and also suppose that ...
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70 views

Are isometries always linear?

Let $E$ be a finite dimensional vector space (over a field of characteristic zero) and $f : E \rightarrow E$ be an isometry fixing 0. Must $f$ be linear in this case ? Note : I am NOT assuming that ...
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291 views

Equivalence of Frobenius norm and trace norm

According to [1], [2] and other related publications, the following holds for any matrix $X$: $$\| ...
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Norms on $\mathbb{Q}$

So with respect to the metric $d(x,y)=|x-y|$ induced by the standard absolute value, the real numbers can be constructed as a completion of $\mathbb{Q}$. With respect to the metric $d_p(x,y)=|x-y|_p$ ...
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Two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on $X$, and a function $f:X\to Y$ Fréchet differentiable with one of the norms but not with the other one?

There is a theorem that if $f : (X,\|\cdot\|_{X1}) \to (Y,\|\cdot\|_{Y1}) $ is Fréchet differentiable, then replacing the norms with some equivalent norms $\|\cdot\|_{X2}$ and $\|\cdot\|_{Y2}$ ...
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104 views

Proving norm equivalence in $W^{1-1/p,p}(\Omega)$

Define for $p\in [1,\infty)$ and $\Omega=(0,1)^N\subset\mathbb{R}^N$ $$W^{1-1/p,p}(\Omega)=\left\{u\in L^p(\Omega): \ \int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy<\infty\right\}$$ ...
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198 views

Prove this equality in a Hilbert Space $H$ with Riesz's Representation Theorem.

For $x \in H$, prove that $\sup_{\|z\| = 1} \langle x , z \rangle= \| x \|$ Here is a quick one. If someone could improve this it would be great Proof By Cauchy Schwarz, $\langle x,z \rangle ...
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The norm of an operator

Let $\rho(x)$ be a weight function in a unit sphere, such that \begin{equation} \begin{array}{l} \displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\ \displaystyle 2. \rho(x)\in ...
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Exercise from textbook about norm

The below diagram included a relevant page and the question 4 of the exercise which i am not sure how to do. Any hint or solution are welcome Attempts: a) I have done it b) i have tried to show that ...
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Is $\rho(f) := \mu(|f| \geq \epsilon), \epsilon >0$ some kind of generalized norm?

Let $\rho(f) := \mu(|f| \geq \epsilon)$ where $\epsilon >0$ is a constant and $\mu$ is a measure and $f$ is a measurable function. Then $f\equiv 0 \rightarrow \rho(f) = 0$ when $|a| \neq 0$, ...
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Confusion about matrix norms

Reading the wiki article I get confused about matrix norms. My question, is it true that $$\lVert Mx \rVert \leq C(\sup_{ij}{M_{ij}}) \lVert x \rVert$$ where $M$ is a matrix and $x$ is a vector and ...
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Minimizing maximum absolute column sum norm of the residual between a matrix and its $k$-rank approximation

Let $X \in \mathbb{R}^{m\times n}$ be a matrix with rank $r$. How can we find the optimal $\tilde{X} \in \mathbb{R}^{m\times n}$ whose rank is $k$ where $k\leq r$ and the reconstruction error in ...
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462 views

How to prove $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$?

I'm trying to show that $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$. A hint would be nice.
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321 views

What is the difference between $L^2$ norm and $\ell^2$ norm?

I can find no precise definitions on the internet for the $L^2$ and $\ell^2$ norms. Certain websites keep switching between the two. Can someone please help me?
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Convergence of a pair linearly independent elements of a vector space

Let $v_1$ and $v_2$ be linearly independent elements of some normed vector space $V$. We have a sequence $(a_n,b_n)$ and are told that $a_nv_1+b_nv_2$ converges to the zero element. Is it the case ...
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Why is $\sqrt{\sum_{i=1}^n |v_i|^2} \leq \sum_{i=1}^n |v_i|$ true?

Sorry if this is very basic but here's a question. Let $\mathbf{v}=(v_1,\ldots, v_n)\in k^n$ where $k=\bar{k}$. Why do we have $$ \sqrt{\sum_{i=1}^n |v_i|^2} \leq \sum_{i=1}^n |v_i|, $$ ...
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586 views

How do you prove the $p$-norm is not a norm in $\mathbb R^n$ when $0<p<1$?

I see that it fails to satisfy the triangle inequality by example but I don't see how to prove this is the case for all $0 < p < 1$. The definition I am using for $p$-norm is $$ \|A\|_p= ...
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257 views

Why do we have the notions of both 'norm' and 'metric'?

In the case of vectors in euclidean space, for instance, we can express one in terms of the other--i.e. length is distance from zero, distance is the length of the vector difference. Does this break ...
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2answers
45 views

Is $u^TAu \geq 0$ true for all symmetric matrices $A$?

we know from the definition of inner product and norm, that $u^Tu$ is always larger than zero, except the case where $u=0$ at which case it is zero. I came across a question that infers that $u^TAu ...
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Prove that $N(\gamma) = 1$ if, and only if, $\gamma$ is a unit in the ring $\mathbb{Z}[\sqrt{n}]$

Prove that $N(\gamma) = 1$ if, and only if, $\gamma$ is a unit in the ring $\mathbb{Z}[\sqrt{n}]$ Where $N$ is the norm function that maps $\gamma = a+b\sqrt{n} \mapsto \left | a^2-nb^2 \right |$ I ...
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Determination of some operator norms

I have to determine the operator norms, the kernels and the images of the following 2 maps: 1) $F_1 :\{x\in C^0([0,10],\mathbb R)|x(0)=0\}\rightarrow C^0([0,10],\mathbb R)$ ...
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131 views

How does one prove $\left\lVert\sum\limits_{n=1}^{\infty}f_n\right\rVert\le \sum\limits_{n=1}^{\infty}\lVert f_n\rVert$?

Can the triangle inequality for norms works for infinite some as well? Hope that someone can give some hints to prove it or give a counterexample. Thanks!
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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 ...
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585 views

$q$-norm $\leq$ $p$-norm [duplicate]

[Ciarlet 1.4-8] If $0 < p < q$, show that $$\left(\sum_{i=1}^n|v_i|^q\right)^{1/q}\ \leq\ \left(\sum_{i=1}^n|v_i|^p\right)^{1/p}$$ Somebody knows how prove that? Thanks in adavance for the ...
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Why does spectral norm equal the largest singular value?

This may be a trivial question yet I was unable to find an answer: $$\left \| A \right \| _2=\sqrt{\lambda_{\text{max}}(A^{^*}A)}=\sigma_{\text{max}}(A)$$ where the spectral norm $\left \| A \right ...
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4answers
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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 ...
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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 ...
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760 views

norm of a linear operator

On http://www.proofwiki.org/wiki/Definition:Norm_(Linear_Transformation) , it is stated that $||Ah|| \leq ||A||||h||$ where $A$ is an operator. Is this a theorem of some sort? If so, how can it be ...
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298 views

Matrix norms involving singular values

The induced 2-norm of a $m \times n$ matrix $A$ is known as $$ ||A||_2 = \underset{||x||=1}{\text{max}} ||Ax||_2 = \sigma_1,$$ where $\sigma_1$ is the largest singular value of $A$. Then, is it ...
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86 views

Does a norm have to map to $\mathbb R$?

Wikipedia defines a norm on a vector space $V$ as a function $p : V \mapsto \mathbb R$. I've seen this defined similarly elsewhere. However, it seems to me that a real codomain isn't always necessary. ...
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2answers
113 views

is product of norms convex?

Is a function of the form $f(x) = \|x\|_1\|x\|_2$ convex in x? I have tried plotting it in wolfram alpha and it appears convex, althought I ahve not been able to show it yet
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2answers
126 views

Proving that $\|x\|_2 \geq \|x\|_1$?

How would you prove that $\|x\|_2 \geq \|x\|_1$, or in other words that $$\sqrt{\int_0^1|f(x)|^2 dx} \geq \int_0^1|f(x)|dx \quad?$$
2
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1answer
129 views

Convergence of $\frac1m(I+A+A^2+\cdots+A^{m-1})$

Let $A$ be an $n\times n$ matrix of nonnegative entries such that $A_{i1}+A_{i2}+\cdots+A_{in}=1$ for all $i\in\{1,2,\ldots,n\}$. What does $A$ have to satisfy so that the sequence ...