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

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Does the norm map respect diamonds of fields extensions?

Suppose we have fields $F_1, F_2$, and we form the compositum $E = F_1 F_2$. We then have a diamond of fields involving $E, F_1, F_2, F_1 \cap F_2$. Is it true that $$x \in F_2 \implies ...
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44 views

Can someone explain why $\| \cdot \|_p, p = \frac{1}{2}$ isn't a norm?

$\| \cdot \|_p, p = \frac{1}{2}$ or "half norm" is not a norm What is a quick way to verify that it is indeed not a norm?
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18 views

Bound of norm of ordered numbers

Suppose we have two sets of real numbers $x$ and $y$ each with $n$ elements. Suppose I order the sets such that $x_{(1)} \geq x_{(2)} \geq x_{(3)} \geq ... \geq x_{(n)}$ and $y_{(1)} \geq y_{(2)} ...
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54 views

In which sense an expansive matrix is expansive

Let $A\in M_n(\mathbb{R})$. $A$ is called an expansive matrix if every eigenvalue of $A$ is strictly bigger than 1. From the name "expansive", one may deduce that (the space of square expansive ...
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1answer
68 views

Proving $\lVert x \rVert_3$ is smaller than euclidean norm

Is there a simple way to show that $\lVert x \rVert_3\leqslant \lVert x \rVert_2$ for vectors in $\mathbb{R}^2$? I've used the result in another problem but can't figure out where to get it from.
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66 views

An identity involving matrix norm and vector norm in relation to a density matrix in physics

This question is self-contained. For those who are interested, it arises from my study of the paper: K Lendi (1987), Evolution matrix in a coherence vector formulation for quantum Markovian master ...
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21 views

Visualising a region involving the $1$-norm

I'm a long-time lurker but first-time poster so I apologise if this post isn't formatted correctly. I'm having trouble visualising the region $\mathcal{R}$ defined as: ...
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62 views

How do you show that the norm for square matrices is submultiplicative?

In these notes: http://www.math.usm.edu/lambers/mat610/sum10/lecture2.pdf It says that square matrices satisfy so called submultiplicative norm $\|AB\| \leq \|A\|\|B\|$. Is it by definition or is ...
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1answer
53 views

Limits of the p-norm

Let $f:[0,1] \rightarrow R$ with $1 \leq f \leq 2$ set $$N(p)=\left( \int_0^1 f^p dx \right)^{\frac{1}{p}} \qquad p \neq 0$$ To find the three limits $\lim_{p\rightarrow \pm \infty} N(p)$ and ...
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Does the inner product $\langle \cdot, \cdot \rangle$ induce any other norms other than the 2 norm?

In the lecture my professor wrote that the standard inner product on $R^n$ is given by $\langle x, y \rangle = x^Ty = \sum\limits_{i=1}^n x_i y_i$ which induces a norm $\sqrt{\langle x,x \rangle} ...
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30 views

To show Banach space with the norm lp(Z)

I am thinking the following task where my proposal is too complicated, I think. It has been difficult for other people to understand and I want to improve it. There are probably some critical steps ...
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28 views

$x^Ty\leq 1, \forall\text{ y with }||y||_2 = 1 \iff ||x||_2\leq 1$

I came across this: This follows from Cauchy-Schwarz inequality: $x^Ty\leq 1, \forall\text{ y with }||y||_2 = 1 \iff ||x||_2\leq 1$ where $x\in\mathbb{R}^n$ When I try to do it myself, this ...
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1answer
82 views

Show strong but not norm convergence on $L^p$

The task: let $1 \le p < \infty$ and $A_k : L^p (\Bbb R) \to L^p (\Bbb R)$ such that $(A_k u) (x) = u(x+\frac 1 k)$. Show that $\| A_k u - u\|_p \to 0$ as $k \to \infty$ (for all $u \in L^p (\Bbb ...
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2answers
130 views

Comparison between the $\ell_1$ and $\ell_2$ norms

I am trying to prove that for $x \in \mathbb{R}^n$ $$\frac{1}{\sqrt{n}}||x||_1 \le ||x||_2 \le ||x||_1$$ So far I have successfully proven that $||x||_2 \le ||x||_1$, which was fairly easy. I took ...
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59 views

Why is the norm ball a square in $\,\mathbb R^2\,$ under $\,l^\infty\,$ norm?

Suppose $\,x = \left(x_1, x_2\right)$, then $\,l^2\,$ norm ball is $\,\left\lbrace x\;\big\vert\;\, \sqrt{\left\lvert x_1 \right\rvert^2 + \left\lvert x_2 \right\rvert^2} \leq 1\right\rbrace$ Easily ...
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39 views

What are the implications of the dual norm of a norm?

I was doing a series of questions proving that the dual norm of $l_p$ is $l_q$, where $p,q$ satisfies $\frac{1}{p} + \frac{1}{q} = 1$. I was able to prove this result but I do not see the point of ...
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28 views

Why does a finite operator norm of a linear transformation imply uniform continuity.

I am following Real Mathematical Analysis by Pugh. I do not understand this simple proof which gives (b). But why is $|Tv - Tv'| \le ||T|||v-v'|$?
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40 views

Is the norm ball a set or the boundary of a set?

Recall normed ball in $R^2$ under different norms is typically intuited as follows But looking at someone of the definition of normed ball it seems that it describes a closed set rather than the ...
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strict inequality with norms $\frac{||y+p|(x+p)-|x+p|(y+p)+|x|y-|y|x|}{|(1+|y|)x-(1+|x|)y|}<2|p|$

We work in $R^n$ and $|x|=\sqrt{x_1^2+...+x_n^2}$. Let $p\in R^n$ different from $0$ and $x,y\in R^n$ with $x\neq y$ then \begin{equation*} ...
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44 views

Prove $\|x\|_1\le \sqrt n \|x\|_2$ [closed]

Prove $$\|x\|_2 \le \|x\|_1 \le \sqrt n\|x\|_2.$$ I already proved the first inequality $\|x\|_2 \le \|x\|_1$. Please help me with the second part: $$\|x\|_1\le \sqrt n \|x\|_2.$$
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106 views

Voronoi diagram boundaries with Manhattan distance

I'm trying to draw a voronoi diagram using the Manhattan distance by hand, and I'm becoming very confused because it appears as though the boundary is an area rather than a line. This seems completely ...
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37 views

Is $\max(|a_{ij}|) \le \| A \|_2$ valid for a complex matrix?

$\max(|a_{ij}|)$ is a maximal magnitude of the elements of a matrix $A$ and $\| \cdot \|_2$ is a 2-norm. I've found this inequality for real matrices in the Matrix Computations, Gene H. Golub and ...
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42 views

Proving an induced operator norm equality:

The induced matrix norm is defined by $$||A|| = \sup_{x \ne 0} \frac {||Ax||}{||x||} $$ Show that $$||A|| = \sup_{||x||=1} ||Ax||$$ A is only assumed to be square -- not anything more, e.g., not ...
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37 views

Question regarding Lipschitz condition $\| f_1 - f_2 \| \leq k \|x_1 - x_2\|$

Consider global Lipschitz condition: $\| f(x_1) - f(x_2) \| \leq k \|x_1 - x_2\|$ We can manipulate it to: $\frac{\| f(x_1) - f(x_2) \|}{\|x_1 - x_2\|} \leq k $ But according to the definition ...
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1answer
47 views

Can the norm of a convex combination of vectors be greater than the norm of their sum?

Can the norm of a convex combination of vectors be greater than the norm of their sum? Or is it always the case that: $|| \sum_i w_i x_i || \leq \sum_i w_i \;|| \sum_i x_i || $ (for positive ...
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Is it true that $\|f\|_p\le c\|f\|_r$ when $r<p$?

Is it true that if $p,r\in [1,\infty]$, $r<p$ then $\exists c>0$ such that $\|f\|_p\le c\|f\|_r$ for all $f\in C_{[0,1]}$? A friend of mine told me that it was false but I haven't been able to ...
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42 views

equality of norms

I have to show that $\| A\|_2=\sqrt{\| A^H\times A\|_2}$. $A$ is a $n\times k$ matrix, $\| \cdot \|_2$ is defined as : $$\| A\|_2=\sup\{ \|Ax\|, \|x\|\leq 1 \}$$ and $\| \cdot \|$ is the ...
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35 views

Scalar equation including $\ell_1$ norm.

My problem is very simple: is there an analytic solution to $$ \mbox{find } \alpha \mbox{ such that } \|u + \alpha v\|_1 = \beta $$ Here $u$, $v$, and $\beta$ are known quantities. Thanks a lot ...
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61 views

Is this a valid norm?

For a positive semidefinite invertible matrix X define the following quantity over the set of symmetric matrices M $$f_X(M) = \sqrt{Trace(XM^2)}$$ Is $f_X(M)$ a valid norm? If yes is it easy to ...
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44 views

Why is $(\|x\|_2)^2$ given by $x^T\!x$ where x is a vector?

Suppose that $x$ is a real number, then $$(\|x\|_2)^2 = x \cdot x = x^2$$ Now suppose $x$ is a real vector, then $$(\|x\|_2)^2 = x^T\! x $$ Why should it be obvious that the multiplication sign ...
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62 views

Showing a norm inequality

Suppose $x \in \mathbb{R}^n$, show $||x||_2 \leq \sqrt{n} ||x||_{\infty}$ attempt: We have $||x||^2_2 = |x_1|^2 + ... + |x_n|^2 $. We know there is some $k$ such that $|x_k| = \max_{1 \leq j \leq n ...
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78 views

Difficulty understanding the proof of equivalence of all norms over $\mathbb R^{n}$

To prove that all norms are equivalent on $\mathbb R^{n}$ , the book I am reading , first takes an arbitrary norm $$|\ \ | \ :\ \mathbb R^{n}\rightarrow\ \mathbb R$$ and then ...
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87 views

In the proof that $L^{1}$ norm and $L^{2}$ norm are equivalent.

To prove that $L^{1}$ norm , denoted by $||\ \ ||_{1}$ and $L^{2}$ norm , denoted by $||\ \ ||_{2}$ are equivalent we have to find constants $C_{1},\ \ C_{2}$ that satisfies ...
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41 views

Differentiation of norm

How do I differentiate the "norm" of $(x-μ)$, with respect to $μ$, where both $x$ and $μ$ are vectors ? How will I start and proceed ? Thank you in advance.
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36 views

Strategy for establishing the triangle inequality of a seminorm

One proof that the $p$-norm $\| x\|_p = (|x_1|^p + \ldots + |x_n|^p)^\frac{1}{p}$ satisfies the triangle inequality exploits the fact that $ x \mapsto |x_1|^p + \ldots + |x_n|^p$ is a convex ...
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38 views

How to find upper bound for the absolute value of the expression?

How to find upper bound for theabsolute value of the expression: $$z = x^T G^T H y$$ where dimensions are $x^T$: $1 \times n_1$; $G^T$: $n_1 \times n$; $H$: $n \times n_2$; $y$: $n_2 \times 1$; ...
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1answer
32 views

Compute the norm of the operation $A$

Suppose that $\left ( a_{ij} \right )_{i,j=1}^{\infty}$ is a matrix satisfying the following condition $$\sum_{i,j=1}^{\infty} \left | a_{ij} \right |^q < \infty$$ where $q>1$. For $x=\left \{ ...
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Conergence of linear functional

I want to determine wether the functional $\varphi_n:\ell^2\to \mathbb{R}$ defined by $$\varphi_n(x)=\frac{1}{n}\sum_{k=1}^n\sqrt k x_k\quad x=(x_1,x_2,\dots)$$ converges in norm, or in weak sense. ...
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Gaussian rationals with rational norm

Looking for information on Gaussian rationals with rational norm. A gaussian rational is a complex number of the form z = p + qi where p and q are rationals. Taking only those that have |z| = ...
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Properties of Linear Transformations

The book I am using is Differential Equations and Dynamical Systems by Lawrence Parko. Seeking to confirm my attempt at proving the following. Use the lemma in this section to show that if $T$ ...
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24 views

Linear Systems and Linear Transformation

I want to confirm my attempt to see if I am on the right track. The question is as follows. Show that the operator norm of a inear transformation $T$ on $\mathbb{R}^n$ satisfies ...
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1answer
59 views

$\operatorname{Im} A = (\operatorname{ker} A^*)^\perp$ [duplicate]

Let $A:\mathbb{R}^m \to \mathbb{R}^n$ be a linear transformation. We know that there is a unique transformation $A^*:\mathbb{R}^n \to \mathbb{R}^m$ such that $$\langle Ax,y\rangle = \langle x,A^*y ...
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33 views

Euclidean norm bound

Consider the variable $x = (x_1,\ldots,x_n)$, where each element $i$ is restricted to live in the interval $\underline{x}_i\le x_i \le \bar{x}_i$. Now consider the norm $\|x\|_2$. Why can't I write ...
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Give an example of a linearly independent sequence $[x_0,x_1,x_2,\dots]$ of vectors in $\ell^{\infty}$ such that $\sum_{n=1}^{\infty} (x_n) =0$

Give an example of a linearly independent sequence $[x_0,x_1,x_2,\dots]$ of vectors in $\ell^{\infty}$ such that $\sum_{n=1}^{\infty} (x_n) =0$. Solution: Let $X_n=[x_0,x_1,\dots]$; define ...
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54 views

staircase length in Whitney's flat norm and Jenny Harrison's natural norm

Can someone provide the complete calculation for the length of a staircase as it converges to a diagonal line in Euclidean space in a sequence in which the number of steps goes to infinity between two ...
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54 views

Is there any inner product on $M_{n \times n}$ inducing this norm?

The set $M_{n \times n}$ is the collection of all $n \times n$ matrices over $\mathbb{R}$. Definition: $\|A\|_2=\sup_{\|u\|_2=1} \|Au\|_2$. Is there any inner product on $M_{n \times n}$ inducing ...
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56 views

The choice of scalar factors in the proof of the Schwarz inequality

In this proof for the Schwarz Inequality, they seemingly arbitrarily choose $r = w\cdot w$ and $s =-(v\cdot w)$. Why did they make these selections? I don't understand where these values for $r$ and ...
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152 views

Upper bound on infinity norm of inverse of a positive definite matrix [closed]

Consider a positive definite matrix, $A$, and the following quantity: \begin{align} \|A^{-1}\|_\infty \end{align} Are there any upper bounds on the above normed term? EDIT: The bound should be in ...
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1answer
47 views

Is the spectral radius of a matrix a convex norm of it?

I am wondering if the spectral radius of a matrix is may be some kind of a norm ($l_{\infty}$-norm?) of it and if that is convex. Any pointers to related ideas would be helpful too.
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Does this property of time dependent sequences have a name?

For $i\in \mathbb{N}$ let $ \chi_i \colon \mathbb{R}^+ \to \mathbb{R}$ be such that for each $t\in \mathbb{R}^+$ we have $\chi_i(t) \in \mathcal{l}_2$ Suppose further that ...