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 to prove a matrix norm inequality?

$P$ is a stochastic matrix i.e. square, non-negative, rows sum to one. Let $\Phi$ be a real matrix of size $n \times k$ with independent columns and $k < n$. Let $\Xi$ be the diagonal matrix with a ...
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42 views

value of the norm of the trace mapping

1) What is the exact value of the norm of the trace mapping ${\rm tr} \colon M_n \to \mathbb{C}$ where we equip $M_n$ with the operator norm $\|A\| = \sup\{\|Ax\| : x\in \ell^2_n \mbox{ with }\|x\|= ...
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1answer
103 views

Is it true that $A \geq B$ implies $\|A\|_2 \geq \|B\|_2$ for $A,B \geq 0$?

All matrices are real and not necessarily symmetric. Denote by $A \geq B$ the condition that $(A-B)$ has eigenvalues with non-negative real parts. Denote by $\| \cdot \|_2$ the $L_2$ matrix norm. Is ...
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111 views

What is the upper bound for this 2-norm

Let $\mathbf{x}$ be the solution to the following problem $$\displaystyle\min_{\mathbf{x}} \|\mathbf{y+Ax}\|_\infty \quad{} \text{subject to} \quad{} \|\mathbf{x}\|_2^2\leq \alpha\|\mathbf{y}\|_2^2$$ ...
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43 views

Find a family of linear inequalities for which the unit ball is a solution set

Find a family of linear inequalities for which the unit ball $A = \{x \in \mathbb R^n \ | \ \|x\|_2 \le 1\}$ is a solution set Would it just be $x_1^2 + ... + x_n^2 \le 1$?
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56 views

Geometric interpretations of $||z||_p = 1$?

Here $z = a + bi$, with $a, b, \in \mathbb{R}$ and $||z||_p = \sqrt[p]{|a|^p + |b|^p}$. With $p = 1$, this is just diamond (square rotated 45 degrees) of side=$\sqrt2$ centered at the origin. With ...
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26 views

Weighted Norm Minimization

I have a minimization problem of the form $min (w_1 \|x\|+w_2\|y\|)$ subject to constraints $A_1x=b_1$, $A_2y=b_2$, $0< l_1 \leq x \leq u_1$ and $0< l_2 \leq y \leq u_2$ where ...
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1answer
88 views

upper bound on this matrix norm

What would be the upper bound on the 2-norm (or any norm) of the following matrix product ? Please consider the smallest upper bound. $\|\left(I+BA^T\right)\left(I+AA^T\right)^{-1}\|< ?$ where A ...
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1answer
44 views

Finding a norm making a subspace dense

Suppose $V$ is a (real or complex) vector space and $W$ is a subspace of $V$. Under what conditions is there a norm on $V$ making $W$ a dense subspace of $V$? That $V$ and $W$ have the same ...
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56 views

Is every regular polygon the unit ball for some norm?

For every regular polygon, is there a norm such that the polygon is it's unit ball centered on 0?
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18 views

Composition of two functions in normed spaces

Let $\Omega_1, \Omega_2 \subset \mathbb{R^n}$ be bounded. The mapping $ F: \Omega_1 \rightarrow \Omega_2 $ shall be bijective, continuously differentiable and such that $||DF(x)||$ and ...
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42 views

Equality condition for convolution's $L^p$ norm.

Suppose that $1< p< \infty$, $f\in L^1(R)$, and $g\in L^p(R)$ and that $\|f*g\|_p=\|f\|_1\|g\|_p$. Show that then either $f=0$ a.e or $g=0$ a.e I have solved for $g=0$ a.e. if $||f||_1>0$ ...
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1answer
49 views

Some Norms on $V=P_n(\mathbb{R})$?

$V=P_n(\mathbb{R})$ be the vector space of all polynomials with degree $\le n$. I need to know Which of the followings are norms on $V.$ $\forall p\in V$ 1.$\|p\|^2=|p(1)|^2+\dots+|p(n+1)|^2$ ...
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51 views

Norm of a mapping which is a linear combination of other mappings

Let $(X, d)$ be a metric space, $(E, || \cdot ||)$ a Banach space, $(AE(X), || \cdot ||)$ - as described below. $AE_0(X) = \{ u : X \rightarrow \mathbb{R} \ : \ u^{-1} (\mathbb{R} \setminus \{0 \}) \ ...
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1answer
115 views

Uniquness and Exisstence of One Theorem

I need a short and nice Proof for Uniqueness and Existence of the following theorem: Suppose (H, <0,0> ) is a Hilbert space, and M is a closed convex set and $x \in H$, then there is a unique ...
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251 views

Prove that the normed vector space $(S_F,\|\cdot\|_1)$ is not Banach.

$S_F$ is the space of real sequences $\mathbf a=(a_n)_{n=1}^{\infty}$ such that every sequence $\mathbf a\in S_F$ is eventually zero. $\|\cdot\|_1$ is the norm defined as $\|\mathbf ...
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1answer
144 views

Show that the following definitions all give norms on $S_F$

$S_F$ is the space of all real sequences $\mathbf a=(a_n)_{n=1}^{\infty}$ such that each sequence $\mathbf a\in S_F$ is eventually zero. Show that the following definitions all give norms on $S_F$, ...
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2answers
148 views

State and prove conditions for $\|x\|_a=\sum_{j=1}^n a_j\lvert x_j\rvert$ to be a norm on $\mathbb R^n$

Let $a_j \in \mathbb R$ for $1\leq j \leq n$. State and prove necessary and sufficient conditions for $\|\cdot\|_a$ to be a norm on $\mathbb R^n$. I know the properties needed for a function to be a ...
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54 views

Defining an unusual norm on $\mathbb{R}^3$

For vectors $\vec v = (v_1,v_2,v_3) \in \mathbb R^3$, does $||\vec v|| = |v_1| + \max\{ |v_2|,|v_3|\}$ define a norm on $\mathbb{R}^3$? I know I need to show positivity, homogeneity and the triangle ...
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1answer
11 views

Calculate 1-norm of a vector using another matrix or vector

Let's say I have a vector a. I would like to construct a matrix or vector b such that if I multiply ...
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1answer
158 views

Prove that $N_2(\mathbf z)=(\sum_{k=1}^n \overline {z_k}z_k)^{1/2}$ is a norm on $\mathbb C^n$

Here $\mathbf z=\{z_k=x_k+iy_k\}_{k=1}^n, \overline{\mathbf z}=\{x_k-iy_k\}_{k=1}^n \in \mathbb C^n$. So far I have proven the following properties required for $N_2(\mathbf z)$ to be a norm: ...
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138 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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79 views

Is $\bigl\|\frac{vv^T}{v^Tv}\bigr\|=1$? For any vector $v\in \mathbb{R}^{n}$

I am stuck while showing that $$\biggl\|\frac{vv^T}{v^Tv}\biggr\|=1,$$ where $v\in \mathbb{R}^n$, and $\|.\|$ is a matrix norm. Here is my steps: I used Frobenius norm: A Frobenius matrix ...
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1answer
120 views

Determine if these two norms are equivalent

Let we have the space $C[a,b]$ (the space of all functions that are continuous on closed interval $[a,b]$). And we have two norms on this space: $$\|X\|_1= \max_{t\in [a,b]} | x(t) |$$ $$\|X\|_2 = ...
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1answer
39 views

Prove $\|\cdot\|_s$ is a norm, and find $m,M>0$ such that $m\|x\|_\infty\leq \|x\|_s\leq M\|x\|_\infty$

Here is my question - Let $\|\cdot\|_s:\mathbb{R}^2\to\mathbb{R}^2$ be defined by: $$\|(x_1,x_2)\|_s=\left\{ \begin{array}{l l} \|(x_1,x_2)\|_2 & \quad \text{$x_1x_2\geq 0$}\\ ...
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1answer
77 views

Convergence as for the norm [duplicate]

If $f_n, f \in L^p, 1\leq p < +\infty$ and $f_n \rightarrow f$ almost everywhere, and $\|f_n\|_p \rightarrow \|f\|_p$, then $f_n\rightarrow f$ as for the norm. Could you give me some hints how to ...
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1answer
41 views

Prove there cannot be an inner product which turns $l^p$ into an inner product space?

For all $1\leq p < \infty, \mbox{ }p$ is not equal to 2, prove there cannot exist an inner product that turns $(X,\|\cdot \|_p)$ into an inner product space; that is, prove that there cannot be ...
3
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1answer
270 views

Norm of Prime Ideal

Show that the norm of a prime ideal in a number field $K$ is a power of some prime number, i.e., if $P$ is a prime ideal in $O_K$ for some number field $K$, then $N_\mathbb{Q}^K(P)=p^n$ for some ...
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1answer
49 views

Does $d(x,y) = \lvert N(x) - N(y)\rvert$ satisfy the triangular inequality?

Let $N(x)$ be the norm of the vector $X$ and efine $$d(x,y) = |N(x) - N(y)|$$ I want to prove that $d(x,y)$ satisfies the triangular inequality. Here is my attempt: $$|N(x) - N(y)| \leq |N(x)| + ...
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1answer
61 views

An inequality about operator norm

Let $H$ be a Hilbert space and $T\in B(H)$, with $T_{i}\rightarrow T$ in strong operator topology. Then can we prove that $\liminf_{i\rightarrow \infty}||T_{i}||\geq ||T||$ ?
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1answer
81 views

Counterexample for $\| (\Phi^\top \Xi \Phi)^\dagger \Phi^\top \Xi P \Phi \|_2 \leq 1$

All matrices are real. Let $P$ be any stochastic matrix (i.e. square, non-negative, rows summing to one). Let $\Xi$ be a diagonal matrix containing on its diagonal any left eigenvector of $P$ ...
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1answer
137 views

Is k-means clustering guaranteed to converge if using Manhattan distance?

The k-means algorithm is an iterative clustering algorithm that partitions the data points into K clusters (with centroids {$\mu_1, ... , \mu_k$}, minimizing the Sum-of-Squared-Error: $$ SSE = ...
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1answer
123 views

a question about symmetric positive definite matrix and norm

If B is $n\times n$ real symmetric positive definite matrix, then $(x,y)=x^TBy$ definites an inner product on $R^n$. How to prove that $||x||=(x^TBx)^{1/2}$ is a norm on $R^n$?
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51 views

How can I prove maximum norm?

Prove that for a matrix $A \in R^{n \times n}$, we have $$\|A\|_{\infty}=\max_{i=1,...,n}\sum_{j=1}^{n}|a_{ij}|.$$ I know that $$\|A\|_{\infty}=\max\frac{\|Ax\|_{\infty}}{\|x\|_{\infty}}$$ such ...
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Inequality between operator norm and Hilbert-Schmidt norm

I have seen the following inequality here but I don't know where I can find a proof for it. Could somebody give me a hint to understand it or guide me to a reference please? $\|AB\|_{HS} \leq ...
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1answer
35 views

Does this funcion define a norm on $\mathbb{C}^n$?

Let $m$ and $n$ be two given positive integers. And, let $f \colon \mathbb{C}^n \to \mathbb{R}$ be defined as follows: $$ f(x_1, x_2, \ldots, x_n) \colon= \left( \sum_{i=1}^n \sqrt[m]{|x_i|} ...
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1answer
32 views

Find $K$ such that $|(x, y)| > K$ implies $(x - 1)^2 + (y + 2)^2 > C+ 4$.

For any 𝐶 ∈ ℝ, find 𝐾 such that |(𝑥, 𝑦)| > 𝐾 ⇒ 𝑥2 + 𝑦2 - 2𝑥 + 4𝑦 + 1 > 𝐶 i.e. (𝑥 - 1)² + (𝑦 + 2)² > 𝐶 + 4 whenever |(𝑥, 𝑦)| > 𝐾 NOTE: 𝐾 is a function of 𝐶 only, and does NOT depend ...
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63 views

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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1answer
23 views

Schikhof's Ultrametric Calculus - Uniquely extending a norm from an integral domain to its quotient field.

This is the problem from Schikhof's Ultrametric Calculus: Let $D$ be an integral domain and $\|\cdot\|:D\to\mathbb{R}$ be a norm. Show that $\|\cdot\|$ may be uniquely extended to a norm on the ...
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perturbation of trace norm

The definition of trace norm is the summation of singularities of that matrix. I need to calculate the trace norm of matrix with the form $$A = I + r$$ where the ...
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1answer
90 views

Quantum fourier transformation Unitary proof.

I've found a bunch of these proofs online but I am having trouble understanding how the norm of the column/row is 1.
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Finding a norm on $ \mathbb{R}^X $ such that the “natural” embedding of a metric space $ X $ in $ \mathbb{R}^X $ becomes an isometry

This question is related to this one: Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set Kuratowski's embedding indeed seems to be the ...
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34 views

Prove the norm inequality.

Exercise. Let $p_1$, $p_2$ be norms on $\Bbb R^n$ with respective unit balls $D_!, D_2$. Prove that $$D_2\subset D_1\iff p_1(x)\le p_2(x)\ \forall x\in\Bbb R^n$$ Can someone please help me this. ...
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29 views

$p$-norm on $\mathbb{R}^n$ question

How I can show that $$\lim_{p \to \infty} \|x\|_{p} = \max\{|x_1|, \; |x_2|, \; \cdot ,\; |x_n|\}$$ if $\mathbb{R}^n$ has the p-norm? $p > 1$ of course. Has anyone done this or know how to? I'm ...
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1answer
102 views

upper bound on a matrix norm

what is the smallest upper bound for the following norm $\|\left(\lambda\ I +A\ A^T\right)^{-1}\|<?$. where, A is a rectangular matrix, $\lambda>0$ is a scalar. (any possible norm)
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78 views

Trace norm identity (in bra-ket notation)

I came across the following identity in a paper: $$ \|\hspace{0.3em}|v\rangle\langle v| - |w\rangle\langle w|\hspace{0.3em}\|_{tr}=2\sqrt{1-|\langle v|w\rangle |^2}$$ where the norm on the left is ...
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22 views

Norm of arbitrary constant

I'm sitting in front of an exercise (basics in quantum mechanics), which wants me to check if the integral of a given function can be normed. One of those functions is the integral of zero from ...
3
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1answer
46 views

Is it true that $\|\text{diag($\pi$)} P\|_2 \leq 1$ for $P$ stochastic and $\pi P = \pi$.

$\| \cdot \|_2 $ is the matrix norm induced by $L_2$. $P$ is any given real square $n \times n$ non-negative matrix with rows summing to one, i.e. $P1 = 1$, where $1$ is the vector of ones. There is ...
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1answer
21 views

When is $\| X \| _\star = \| F X \| _2$ submultiplicative?

All matrices are real. By $\| \cdot \|_2$ denote the matrix norm induced by $L_2$. Assume $F$ is an invertible matrix. Consider the norm $\| X \| _\star = \| F X \| _2$. What is the condition on ...
4
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3answers
111 views

Prove that $0 < x < y$ implies $\|x\| < \|y\|$ for any norm.

All vectors are real. Prove that $0 < x < y$ (element-wise) implies $\|x\| < \|y\|$ for any norm. This is probably very basic, but I don't seem to get the hang of it. Edit: it turns out this ...