# Tagged Questions

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $R^n$?
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 ...