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Questions tagged [norm]

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

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$\alpha N_1-\beta N_2$ is a norm where $N_2<N_1$

Given $N_2(x)\le N_1(x)$ norms on some vector space over $F$ with equality only holding iff $x=0$ and $N_1\ne\delta N_2$, is it possible that $\alpha N_1-\beta N_2$ is a norm ($\alpha>\beta>0$) ...
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Weak-continuity of an operator on the vector space of measures

Hello and thanks in advance for your time. Let $\mathcal{S} \subset \mathbb{R}^d$ for some integer $d$ and let $M(\mathcal{S})$ be the space of finite signed Borel measures on $\mathcal{S}$. We also ...
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38 views

Cauchy Schwarz inequality with 1 norm

Here is the argument I am making By Holder's inequality, we have for $\frac{1}{p} +\frac{1}{p^*} = 1$ $$\langle A, B\rangle \leq ||A||_p||B||_{p^*}$$ The Schatten p-norms also obey $||A||_p \geq ||...
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Uniform boundedness of the operator in Grand Lebesgue Space

I want to show uniform boundedness of the projector operator in Grand Lebesgue space. Note that, the norm of $f(x)$ in ${ L }^{ p) }\left( -\pi ;\pi \right) $ is $${ \left\| f \right\| }_{ p) }=\...
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Question about an inequality about geometric mean

I encountered this inequality when reading some paper, however, why it holds does not seem very obvious to me: Consider nonnegative real numbers $a_1, ..., a_n, b_1, ..., b_n$, and $b_i \leq \frac{...
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Normed space $C^2[0,1]$ with norm $\lVert f\rVert:=\max_{t\in[0,1]}\{\lvert f(t)\rvert+\lvert f''(t)\rvert\}$ is Banach space

The problem is as follows: I want to show that the normed space $C^2[0,1]$ with norm defined as $$\lVert f\rVert:=\max_{t\in[0,1]}\{\lvert f(t)\rvert+\lvert f''(t)\rvert\}$$ is a Banach space (and I ...
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Importance of equivalence of norms

I am doing a course in Convex Optimization where I learned about the equivalence of norms, but there was no mention of its importance or about the scenarios where it can come in handy. The definition ...
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Can someone guide if I'm proceeding correctly with solving the below problem?

$A\in R^{n\times d}$ : consider it to be full rank. $x \in R^{d}$ $A = U \Sigma V^{T}$ : SVD decomposition of A: I'm suppose to find the value of below terms $\sup_x \frac{|| \Sigma V^{T}x||_{\infty}...
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Upper bound on expected norm of subgaussian random matrix

Let $A \in \cal{M}_{n \times m}(\Bbb{R})$ be a random matrix with IID subgaussian entries with variance proxy $\sigma^2$. Show that $E[||A||_{op}] \le c \sigma \sqrt{m+n}$ for a constant $c$ to be ...
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To show a norm is finer than other norm

Let $V$ and $W$ be two Banach spaces and let $T \in L(V,W)$ be such that $R(T)$ is close and dim $N(T)< \infty$.Let |.| denote another norm in $V$ with $|x|\leq M||x||_V$ for all $x\in V$.Prove ...
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What vector x will maximize the norm of $\|Ax\|_2 / \|x\|_2$ (norm 2)

I know for norm one vector $x$ should be a basis vector. Where one is in the column of matrix $A$. and for infinity norm $x$ should have elements $-1$ for negative values of the maximum row and $+1$ ...
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How to determine the supremum of a set with respect of all vectors of norm 1?

I'm having some troubles regarding the definitions that use the supremum of a set for all vectors with a specific norm. At the moment the case in question is the operator norm, defined as $\|T\|_{X'} ...
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Using a Euclidean norm to bound a $k$-tuple

This does not look too complicated, but I've been stuck here for several hours. My question is to prove that $||(h, \cdots, h)||\leq ||h||^{k}$, where $||\cdot||$ is the euclidean norm, and $(h,\cdots,...
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Orthonormal Basis In $l_1$ Norm

If I am working on finite dimensional innerproduct space $\mathbb R^m$, and $\{v_{1},\dots,v_{m}\}$ be a orthonormal basis for $\mathbb R^m$, can I say that they are orthonormal in the sense of $l_1$ ...
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Matrix norm if we don't specify the vector norm

Is it possible to compute the norm $$ ||A||=\sup_{X\in \mathbb{R}^n\setminus \{0\}} \frac{\lVert AX\rVert}{\lVert X\rVert} $$ where $A$ is the matrice defined by $AX=(x_1-x_n,x_2-x_1,\ldots,x_n-x_{n-1}...
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When will a prime element of $\Bbb{Z}[(\sqrt{5}-1)/2]$ have field norm equal to a rational prime?

Consider the integer ring of $\mathbb{Q}[\sqrt{5}]$, i.e. $\mathbb{Z}[(\sqrt{5}-1)/2]$. Then if $N(x)$ denotes the field norm of $x\in\mathbb{Z}[(\sqrt{5}-1)/2]$, then $N(x) = p$ for a rational prime $...
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norm of difference of similar matrice

Let $a,b \in C^n$; $A, B\in C^{n\times n}$. If $A$ and $B$ are similar matrices, i.e. there exists nonsingular $S\in C^{n\times n}$ such that $B=S^{-1}AS$, is it possible to proof an inequality in the ...
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First order optimality condition on orthogonal projection

I don't understand what is the first order optimality condition, and why do we need it. I think the idea is to prove that $f'(x) = 0$ Also, regarding orthogonal projection, I don't understand the ...
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Bounding the dot product of two planar unit vectors.

Does there exist a continuous, monotone increasing function $f\colon[0,2]\to [0,1]$, satisfying $f(0)=0$ and $f(1)=1$, such that for all vectors $(a_1,b_1),(a_2,b_2)\in \mathbb{R}^2$ of unit length, i....
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solving vector equations involving the L2 (or any other) vector norm

My question is very simple. How can we solve a vector equation involving the L2 norm, for example. $$a + 2\frac{x}{\Vert x \Vert _2}=0$$ Even if a write it this way: $$a + 2\frac{x}{\sqrt{\sum_i ...
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Intuition of Euclidean norm

I know the following : A norm is Euclidean iff it respects the parallelogram law. The problem is that the parallelogram law is not very intuitive geometrically. So I am wondering if there is a ...
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Using covariance function to normalise a function

In the statistical inversion setting it is common to adopt a Gaussian Process (GP) Prior with a Gaussian kernel to preferentially treat smooth parameter fields. With covariance matrix, $$S_{ij}=a \; ...
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35 views

What does $\|u\|^2_2$ mean?

Given a vector $u = (x, y, z)$ what is $\|u\|_2^2$ ?
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Does $||x||\leq ||\overline{x}||$ and $||y||\leq ||\overline{y}||$ imply $||x-y||\leq ||\overline{x}-\overline{y}||$?

I've been working with norms for quite a bit now, and I have started to ponder whether $$||x||\leq ||\overline{x}||\text{ and }||y||\leq ||\overline{y}||\Rightarrow||x-y||\leq ||\overline{x}-\...
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Inner products and norms on space of affine functions

Can someone wrap up the situation of norms and inner products of affine functions and how to integrate according to those norms? 1) Let $V$, $W$ be normed vector spaces. Then, the set of continuous ...
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Is the following statement about matrix norms of matrix products correct?

Let $A$ be a real-valued, square matrix and define its 2-norm as: $$||A||_2 = \sqrt{\max_i\lambda_i(AA^T)}$$ where $\lambda_i(AA^T)$ denotes the $i^{th}$ eigenvalue of the product $AA^T$. Now ...
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Intuition on norm of quotient space

Theorem. Let $(X,\| \cdot \|$) be a normed space. Then $$ p(x+U) = \inf_{z \in U} \|z-x \|$$ defines a semi-norm on $X/U$ with $p(x+U) \leq \|x \|$. a) If $U$ is closed, then $p$ is a norm. b) If $U$...
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Why is the norm $1$ of matrix $A$ is equal to the maximum sum of column

first, I know that there exists a similar question to mine which is in here, and it is actually very well explained. However, there is just one part that I do not understand. That is the conclusion. ...
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Proving the infinity norm is equal to the maximum value of the vector

We know that . I am trying to figure out how to prove when p goes to infinity then the norm represent the maximum value of the vector
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Gauge of an absorbing set

We know that in a normed linear space the gauge of the unit closed ball is the corresponding norm. Is it possible to have a nonconvex absorbibg set in a vector space whose gauge is a norm. We know ...
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do we have $:\left\Vert Q^{n+1}x\right\Vert \leq \varepsilon \left\Vert Q^{n}x\right\Vert $ for all $x\in \mathcal{H}$ for a quasi-nilpotent operator?

Let $\mathcal{H}$ be a Hilbert space and let $Q$ be a bounded quasi-nilpotent operator on $\mathcal{H}$. I'm trying to prove that for every$\ \varepsilon >0,$ there is some $n\in %TCIMACRO{\U{...
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Characterization of a rotund space

A normed linear space $X$ is said to be rotund if for all $x,y\in X$ with $\|x\|=1=\|y\|$, $\|x+y\|<2$. I want to prove that a normed linear space $X$ is rotund iff the function $\varphi:X\to \...
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I have written up a proof on why norms on $\mathbb{R}^n$ are equivalent

Just wanted to share my proof with you smart people to have some feedback and to share each other's ideas. Some disclaimers: this is my first post, English is not my first language, and I know that ...
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48 views

Norm estimate for a product of two orthogonal projectors

Let $H$ denote a Hilbert space. Consider two orthogonal projectors $\,P,Q\in\mathscr L(H)\,$ such that $H=\operatorname{Im}P\oplus\operatorname{Im}Q\,,$ that is both $\,\operatorname{Im}Q\,$ and $\,\...
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Do sub-stochastic matrices reduce the 1-norm of vectors?

It is well-known that "stochastic matrices preserve the 1-norm of vectors, $||\mathbf{A}\vec{v}||_1=||\vec{v}||_1$". I am thus wondering whether sub-stochastic matrices (a square matrix $\mathbf{Q}$ ...
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A maximization problem in the paper “Maximum Ratio Transmission”.

On page 1459 in [1], there is a maximization problem: $$ \max_{\mathbf{g}} \gamma, \tag{1} $$ where $\mathbf{g} = [g_1, ..., g_L]$ is an $1 \times L$ complex vector. $L$ is a positive integer. $$ \...
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1answer
9 views

Zero linear transformation in limits with Euclidean norm

This does not seem too difficult, but I've been stuck here for a while. Could someone give me a hint? Question: Let $A$ be a linear transformation on $R^n$. $v$ is a vector in $R^n$. Prove that if $\...
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47 views

A “Crookedness criterion” for a pair of orthogonal projectors?

If $P$ is an orthogonal projector on a Hilbert space $H$, then $\,\operatorname{im}P=(\ker P)^\perp\subset H\,$ is a closed subspace, also called the support of $P$. And vice versa: Every closed ...
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Sum of Fourier coefficients less than the norm of the respective vector

I have the following question to complete. Let $X$ be an inner product space. Let $(e_{j})_{j\geq1}$ be an orthonormal sequence in $X$. Show that, \begin{align} \sum_{j=1}^{\infty}|(x|e_{j})(y|e_{j})|...
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2-norm & infinity norms of a system are dependent or independent of delay?

We are given some transfer function $G(s)$ And the 2-norm and $\infty$ norms are given by $$ {||G||}_2 = (\frac{1}{2\pi}\int_{-\infty}^{\infty} |G(j\omega)|^{2} \text{d}\omega)^{1/2} $$ $${||G||}_\...
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How to properly estimate a Schwartz function

For every Schwartz function $\varphi \in S(\mathbb{R}^{n})$ there exists a constant $c_{\beta, k}$ such that one can estimate the Schwartz function by $$|\partial^{\beta}\varphi(x)| \leq \frac{c_{\...
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Total variation of a measure in a topological dual of the space of continuous functions

I read the definition of total variation of a measure in the following link https://www.encyclopediaofmath.org/index.php/Signed_measure And then I have a question: If $\mu$ is a vector measure in $...
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Maximum number of unit vectors with bounded pairwise dot products

Let $n\in\mathbb{Z}^+$ and $t\in(0,\tfrac{1}{2})$. Upper bound $m$ such that there exist $v_1,...,v_m\in\mathbb{R}^n$ with $\|v_i\|=1$ and $|v_i\cdot v_j|\le t$ for all $i\neq j$. If $t=0$, $v_i\cdot ...
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What would be the corresponding $\text{matrix norm}$ on $\mathbb{R}^3$

We have a norm on $\mathbb{R}^3$ defined by $\vert (u,v,w)\vert=\max\{|u|,|v|+|w|\}$. This is vector norm on $\mathbb{R}^3$. What would be the corresponding $\text{matrix norm}$ on $\mathbb{R}^3$ ...
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27 views

Inner product induced by the polarisation identitiy

I am required to show that given a Banach space $(X,\|\cdot\|)$ on $\mathbb{R}$ that if the parallelogram identity holds then $X$ is a Hilbert space with the inner product given by the polarisation ...
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32 views

Prove a function is a norm?

I want to show that $${||x||}_{\infty} = \max{\{|x_1|,|x_1|,...,|x_n|\}}$$ is a norm. Where the properties of a norm are ${||x||} \ge 0$ for all $x$ in${\mathbb R}^n$ ${||x||} = 0$ when $x=0$ ${||ax|...
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Prove $|tr(PAP') - tr(PBP')| \leq ||A-B||_{\infty} ||P'||_1^2$

How can I show the following identity? $|\text{tr}(PAP') - \text{tr}(PBP')| \leq ||A-B||_{\infty} ||P'||_1^2$
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Norm of a set of vectors with respect to a quadratic form

I've got a problem that I'm struggling to put into a form that I can analyze. Suppose I have a quadratic form $f(x,y)=ax^2+2bxy+cy^2 = \mathbf{u}\mathbf{A}\mathbf{u}^T$ for $\mathbf{u} = \begin{...
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23 views

For metric space $C^0([0,1])$, a sequence of functions ${x^n}$ is Cauchy in $||-||_1$ norm but not in infinite norm

In my text book, it is stated that for a sequence of functions ${x^n}$ in a metric space $C^0([0,1])$, it is Cauchy in $||-||_1$ norm but not in $||-||_\infty$ norm because that would lead to a ...
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Unique multiplicative quadratic form on quaternion algebras

I want to prove, that the only multiplicative quadratic form $Q$ (so $Q(xy)=Q(x)Q(y) \forall x,y$) on a quaternion algebra $\Big(\dfrac{a,b}{F}\Big)$ is the norm $\mathrm{Nr}$, which is isometric to ...