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

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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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121 views

Geometric Sums in Banach Algebra

Let $E$ be a Banach Algebra with identity, and $v\in E$, so that $||v|| < 1$. The geometric series $w = \sum_{k=0}^\infty v^k$ converges in the norm. I can show that $||w|| \le ...
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185 views

How is $L_{2}$ Minkowski norm different from $L^{2}$ norm?

I am reading the book Multidimensional Particle Swarm Optimization for Machine Learning and Pattern Recognition. They use $L_{2}$ Minkowski norm (Euclidean) as the distance metric in the feature space ...
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79 views

Upper bound for norm of Hilbert space operator

It is a standard result that for a bounded self-adjoint operator $T$ on a complex Hilbert space $H$, we have $||T||=\sup_{||x||=1}|\langle Tx,x\rangle|:=M$. It seems that for any bounded operator on ...
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62 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 ...
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71 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 ...
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220 views

Two proofs concerning Hölder's inequality

I am studying functional analysis and I have come across two statements which can be proven by using Hölder's inequality, but I don't know how/why. Reminder: let $a\in l^p$ and $b\in l^q$, then: $$ ...
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291 views

An inequality involving the norms of symmetric positive definite matrices

Given A and B two real symmetric positive definite matrices is it true that, for some norm $\|.\|$, this inequality holds $$ \|AB-I\| \leq \|A^2B^2-I\| \qquad ? $$
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88 views

Bounding the $l_1$ norm of a vector

Let $x$ be real vector with $\|x\|_1=x_1+\ldots +x_{2n}$. How to bound from above $(x_1+\ldots+x_n)(x_{n+1}+\ldots+x_{2n})$ by $l_2$ norm of the vector $x$. Of course, using $\|x\|\leq\sqrt ...
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147 views

The trace norm cannot be increased by composing with a unitary operator

$\newcommand{\tr}{\operatorname{tr}}$ I was reading a proof for the statement $|\tr(US)|\leq |\tr(S)|$, for every endomorphism $S$ on a complex vector space $H$ and every unitary operator $U$ on the ...
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162 views

Proof that $\|\cdot\|_B$ defines a norm

Can you tell me whether my (partial) proof that $\|v\|_B := \inf \{\lambda > 0 \mid \frac{1}{\lambda} v \in B \}$ where $\varnothing \neq B \subset \mathbb R^d$ is open, bounded, $B = -B$ and ...
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62 views

Proof that the Euclidean norm is indeed a norm

I apologize beforehand for this question. Its embarrassing I know. Anyway, here we go. Recall: $$ \| x \|^2_{\mathbb{R}^2} = \sum^{n}_{i = 1} x^2_{i}$$ How do we prove its a norm? Well if its a ...
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50 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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132 views

Why is semi-norm special?

One difference between semi-norm and norm is: "It is possible for ||v|| = 0 for nonzero v, ||.|| being semi-norm" I see some papers, and they use semi-norm directly. Why is semi-norm better than ...
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1answer
139 views

Prove $|\det A| \leq \prod_{j=1}^n ||a_j||$

Let's say A is a square n by n matrix. ||$x$||=$x^T x$ and x is a real n-column norm. How would you show this? I tried to use the QR factorization here in showing that ||$a_j$||=||$r_j$||, but ...
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34 views

Spectral Norm of $2\times 2$ symmetric matrix

Consider a $2\times 2$ symmetric matrix, in this case, is there some closed formula for its spectral norm ? By spectral norm I mean the induced 2-norm, there is a definition here. Thanks.
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107 views

Question about sup norm

Let $x \in \mathbb{R}^n$. Define $|x| = \max\{ |x_1|,...,|x_n|\} $. I want to show that this is a norm on $R^n$. This is my reasoning. First, notice $$ |x| = \max\{ |x_i| \} \geq |x_i| \; \forall i ...
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217 views

Prove that the given distance function is a norm

Let the vector space $X=K^3$. For $x=(\alpha_{1}, \alpha_{2}, \alpha_{3}) \in X$, we define $||x||= [(|\alpha_{1}|^2+|\alpha_{2}|^3)^\frac{3}{2} + |\alpha_{3}|^3]^\frac{1}{3}$ Proof that $||·||$ is ...
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208 views

Norm of random vector plus constant

Suppose that $w$ is a multivariate standard normal vector and $c$ a real vector of the same size. I know that for positive x $$P(||w+c||^2\geq x)\ \geq \ P(||w||^2\geq x)$$ but I cannot prove it. We ...
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52 views

what is the limit of $l_p$ at p=0?

The p-norm is defined as: $$ \ \|x\|_p=\left(|x_1|^p+|x_2|^p+\dotsb+|x_n|^p\right)^{\frac{1}{p}} $$ When $p<1$, this is no longer a "norm" because it violates the triangle inequality (- it is ...
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206 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 ...
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64 views

Approximation of matrix in 2-norm

The question is the following: Given a matrix $A$ with rank $k$, we are looking for a matrix $B$ of rank $j$, where $j<k$ such that $\|A-B\|_2$ is minimal. My idea was to choose, if $A=P ...
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187 views

Can the $0$-norm represent determinism?

In Scott Aaronson's Quantum Computing since Democritus, he presents classical probability theory as based on the $1$-norm, and QM as based on the $2$-norm. Call $\{v_1,\ldots,v_N\}$ a unit vector ...
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43 views

Do the pth powers of $p$-norms define the same partial ordering on the set of all probability distributions for all $p>1$?

Consider the $p$-th power of the Schatten $p$-norm $||q||_p$ of a probability distribution $q$ , ie, the function $\sum_j q_j^p$, where $\sum_j q_j = 1$ and $q_j \geq 0$. For fixed $q$ and $p>1$ ...
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124 views

Is this a matrix norm?

In wikipedia, the operator norm of a matrix is given by (assume: real, $n$-dimensional) $$ ||A||= \max \left\{ \frac{|Ax|}{|x|}:x \in \mathbb{R}^n, x\neq 0 \right\}$$ (I'm not sure why it is not a ...
3
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1answer
139 views

Operator norm and spectrum

Let $L$ be a bounded linear operator on a Hilbert space. Without assuming finite dimensions, can we express the operator norm of $L$ in terms of the spectrum of the positive operator $L^{\dagger}L$? ...
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351 views

Matrix norm inequality implying eigenvector norm inequality

For a matrix $A$ let $\|A\|$ be the norm given by $\|A\|=\sup_{v \neq 0}\frac{\|Av\|}{\|v\|}$ where $\|v\|$ is the Euclidian norm on the vector $v$. Suppose we have matrices $M$ and $S$ with leading ...
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137 views

Bizarre formula for arc length

I'm reading a book on computer science/math and I found this formula for arc lengths that I've not been able to decipher: $$\left|\int_p^q\left\| {df(x)\over dx} \right\| dx\right|$$ where $\lVert ...
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441 views

Is this a norm? (triangle inequality for weighted maximum norm)

I've been trying to prove that the following is a norm, but wasn't successful. I also cannot find a counterexample. So help is greatly appreciated. Let $x \in \mathbb{R}^N, \ w_i \in \mathbb{R}_+,\ ...
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756 views

Proof that every normed vector space is a topological vector space

The topology induced by the norm of a normed vector space is such that the space is a topological vector space. Can you tell me if my proof is correct? Of course we have to show that addition and ...
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How to prove that a unitary matrix with other conditions is diagonal?

Let $A$ be an $n$ by $n$ complex matrix. The singular value decomposition of $A$ gives: $A=U\Sigma V^*$, where $U$ and $V$ are unitary matrices, and $\Sigma$ is a diagonal matrix that contains the ...
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1answer
41 views

Relation between Frobenius norm and eigenvalues

I'm considering a stochastic multivariate process, the stability of which implies that all eigenvalues $\lambda_i$, $i = \overline{1,n}$ of a certain square real-valued matrix $A$ lie within the ...
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1answer
27 views

involution on $\mathbb F_p((t))$

Recently I learned here interesting facts about $\mathbb F_p((t))$, the field of the formal Laurent series with coefficients in $\mathbb F_p$. I looked up quite a few papers on local fields but I ...
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62 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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41 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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39 views

Norm on $\mathbb R^n$ with given unit ball

Consider a finite subset $S$ of $\mathbb Z^n$ such that $-s\in S$ whenever $s\in S$ and $S$ generates $\mathbb Z^n$. What is a norm on $\mathbb R^n$ whose unit ball is precisely the convex hull of ...
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43 views

Operator norm and Hilbert Schmidt norm

I'm looking for a proof of \begin{equation} ||T||\leq ||T||_{HS}, \end{equation} for which it is sufficient to show \begin{equation} ||Tx|| \leq ||x|| \cdot ||T||_{HS} \forall x\in H, x\not=0 ...
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1answer
41 views

Is the norm of the average $\le$ the norm of the max?

Given $\pmb X \in \mathcal{R}^p$, denote the elements of $\pmb X$ as $\pmb x_i$ for $i= 1, \dots, n$. Denote the $t(\pmb X)$ as the average of $\pmb X$ \begin{equation} \pmb t(\pmb X) = \frac 1 n ...
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48 views

Is taking the Euclidean norm of multiple Euclidean norms equivalent to taking the Frobenius norm?

I'm just a programmer venturing into the world of norms (is that even a thing?) here, and am wondering if two formulas are equivalent. Please forgive my ignorance! Suppose we have a $10\times3$ ...
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Asymptotics of Gelfand's formula

In the following paper, it is stated that for for any matrix norm, $n \in \mathbb{N}$ and $A \in \mathbb{C}^{d \times d}$, the following holds: $\rho(A) \ge \gamma^{(1+\ln n)/n}\|A^{n}\|^{1/n}$ for ...
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160 views

Metric induced by a norm - what conditions should this metric meet?

$(I)$ I've been browsing some problems concerning metrics not induced by norms, and I've found a comment that said that such a metric should be a concave monotone function. Here is the post I'm ...
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90 views

What norm makes $C^\infty[a,b]$ a complete space?

I have been searching for some common norms used on vector spaces of functions but I am not having any luck finding what the most common norm is to use on $C^\infty[a,b]$ More specifically I would ...
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148 views

Geometric Mean limit of $\ell_p$ norm of sums

My analysis professor introduced the $\ell_p$ norm to our class as: \begin{align} \| x \|_p = \left(\frac{1}{n}\sum_{j=1}^{n} |x_j|^p\right)^{1/p}. \end{align} We are asked to prove the following: ...
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1answer
104 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 ...
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1answer
57 views

Need help to simplify an equation

I am computing an error estimate where at the end I got the following term $\|X_{k} - G\|\leq (q^{2^{k+2}} + q^{2^{k+3}}+ q^{2^{k+4}}....)q^{-3}\|Y_{0}\| + q^{2^{k+1}}.q^{-2}\|X_0\|$ , where $X_k$ ...
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88 views

About the convexity of Ky Fan's norm

As we know, the Ky Fan norm is convex, and so is the Ky Fan k-norm. My question is, does this imply that the difference between them is a non-convex function, since it results from "difference between ...
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477 views

How to show the unit ball of the dual norm is also polytope?

Assuming the norm's unit ball is a convex polytope. How can one show that the unit ball of the dual's norm is convex polytope and/or polytope ?
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257 views

Matrix proof using norms

I have a linear algebra question I need help with. Let $A$ be an $m\times m$ matrix with $\|A\|_2 < 1$ where $\|A\|_2$ is the $2$-norm of $A$. Show that $I - A$ is invertible where $I$ is the ...
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462 views

Proof of a norm

Prove that: $ \|\cdot\|_1:\mathbb{R}^n \rightarrow \mathbb{R};\vec{x} \mapsto \sum_{j=1}^n |x_j| $ is a norm defined on the vector space $\mathbb{R}^n$. 1) Zero vector: $\sum_{j=1}^n |x_j| = |x|_1 ...
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1answer
451 views

Minimize distance between 2 functions

Just so you know, this is a homework question, and I basically need help with the steps to solve this problem. I understand what it's asking; however, my attempts haven't worked out, and I'm probably ...