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

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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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40 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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1answer
12 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
20 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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17 views

matrix two norm derivative with respect to X

What would be the result of the following derivative in terms of X? $\frac{d \|X\|_2}{d~ X}=?$
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14 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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28 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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12 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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28 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
23 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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19 views

Norm Calculation Problem

I received this problem on an old homework assignment as extra credit, the period for getting credit is long passed but I'm frustrated that I can't even seem to know where to begin this problem. Let ...
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22 views

Norm in $C(X,\Bbb{R})$

Let $X\subset\Bbb{R}$ a compact set and $f\in C(X,\Bbb{R})$. Define $$\|f\|_{\infty}=\sup A_f$$ with $A_f=\{|f(x)|\in \Bbb{R};x\in X\}$. Then $\|f\|_{\infty}=|f(x_0)|$, for some $x_0 \in X$, since ...
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50 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
52 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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2answers
149 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
96 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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105 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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49 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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63 views

Norm definition [closed]

I need a help with this exercise. I should show that the below mentioned expressions define norms on $\mathbb{R}^n$ with $\underline{x}=[x_1, x_2,\ldots, x_n]^T$: $$ \|\underline{x}\|_1=\sum_{i=1}^n ...
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1answer
5 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
118 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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62 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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41 views

Is $\biggl\|\frac{vv^T}{v^Tv}\biggr\|=1$? 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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85 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
30 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
63 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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26 views

Inequality about the $L_2$ norm of stochastic matrices.

Let $P$ be a $n \times n$ stochastic matrix i.e. square, non-negative, rows sum to one. Let $\Phi$ be any given real matrix of size $n \times k$. We can assume $\Phi$ has independent columns and $k ...
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1answer
23 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 ...
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49 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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39 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
50 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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the number of non-zero integral ideals of norm m in a ring of integers [closed]

How to prove that the number of non-zero integral ideals of norm m in a ring of integers of a number field with degree n is less than or equal to the number of n-dim vectors of n positive integer ...
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1answer
75 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
31 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
72 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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37 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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7 views

How to estimate impact on Eigenvalues of a matrix with small entries

I have a diagonal matrix $D$ and a symmetric matrix $M$ (both $\in\mathbb{R}^{n\times n},n\in\mathbb{N}$) with $M_{ij}\ll\min(\{D_{ll}\})\ \forall i,j,l$. Now I want to compute the eigenvalues of ...
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33 views

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
27 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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28 views

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

For any 𝐢 ∈ ℝ, find 𝐾 such that |(π‘₯, 𝑦)| > 𝐾 β‡’ π‘₯2 + 𝑦2 - 2π‘₯ + 4𝑦 + 1 > 𝐢 i.e. (π‘₯ - 1)Β² + (𝑦 + 2)Β² > 𝐢 + 4 whenever |(π‘₯, 𝑦)| > 𝐾 NOTE: 𝐾 is a function of 𝐢 only, and does NOT depend ...
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14 views

Unit Function of $L^1$ norm

Find the unit function that is a constant multiple of the functions $f(x)=x-1/3$ with respect to the $L^1$ norm on $[0,1]$. I've tried this by using $u(x)=f(x)/||f(x)||$, but keep getting the wrong ...
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45 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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13 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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25 views

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
23 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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3answers
29 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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16 views

H norm of delayed transfer function

Compute the $ H_{\infty } $ and $ H_{2 } $ norm of transfer function G(s) based on the real parameter "a". $$G(s)=\frac{1-e^{-as}}{s}$$
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23 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
57 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)