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

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How can I prove that $\int_{a}^{b} |f(x)|dx$ defines norm on $C([a,b])$?

I need to show that $$\parallel f \parallel = \int_{a}^{b} |f(x)|dx$$ is a norm on $C([a,b])$. I need to show that $\|f\|$ meets the properties of a norm: positive distance, if all elements ...
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12 views

Show that $\lVert A \rVert_T := \lVert T^{-1} A T \rVert$ is a subordinate (induced) norm

I saw the following claim in many places without proof: Given an induced norm $\lVert \cdot \rVert$, $$ \lVert A \rVert_T := \lVert T^{-1} A T \rVert $$ is also an induced norm. All of the texts I ...
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53 views

Matrix log norm

How is it that the matrix log norm: $$\lim_{\epsilon\to 0}={||I+\epsilon A||-1\over \epsilon}$$ is equal to $$\max\left( \lambda \left({A+A^T\over 2}\right)\right)$$ (the biggest eigenvalue)
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49 views

Euclidean norm gives length even in $>3$ dimensions?

In $1,2,3$ dimensions I can simply make triangles and see that Euclidean norm gives me the distance between two points (i.e. the length of the vector from one point to the other). In higher ...
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85 views

When is the Frobenius norm equal to the spectral radius?

I know that the spectral radius is $\rho(A) = max |\lambda_l| = \S_{max}^2$ and that the Frobenius norm is $||A|| = \sqrt{tr(A^*A)} = (\sum_{k}S_k^2)^{1/2}$, which means I want to find the matrix A ...
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25 views

Why, when $m < n$, does the vector space $S$ of vectors $x ∈ \mathbb{R}^n$ such that $a_i · x = 0$ always contains a nonzero vector?

Fix vectors $a_1, . . . , a_m ∈ \mathbb{R}^n$, and let S be the set of vectors $x ∈ \mathbb{R}^n$ such that $a_i · x = 0$ for all $i$. Now I know that $S$ is a vector space. Why is it that when $m ...
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32 views

Does $x \cdot y = 0$ imply that $||x+y||_\infty = ||x||_\infty + ||y||_\infty $?

If x and y are orthogonal vectors and we define $||x||_\infty = $ max$_{j = 0 ... 1} |x_j|$, is it possible to express $||x+y||_\infty$ in terms of $||x||_\infty$ and $||y||_\infty$ ? So I get that ...
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67 views

Does $\mathbf x\cdot \mathbf y = 0$ imply that $\lVert x+y\rVert_1 = \lVert x\rVert_1 + \lVert y\rVert_1$?

If x and y are orthogonal vectors and we define $\lVert x\rVert_1 =\sum^{n}_{j=1} |x_j|$, is it possible to express $\lVert x+y\rVert_1$ in terms of $\lVert x\rVert_1$ and $\lVert y\rVert_1$ ? So I ...
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48 views

Why is $|x · y| ≤ ||x||_1||y||_∞$?

So let $||x||_∞ := $ max $_{j=1,...,n}|x_j|$ and by Cauchy-Schwarz, $|x · y| ≤ ||x||_2||y||_2$ . Why then does $|x · y| ≤ ||x||_1||y||_∞$ ? I'm not sure how to show this.
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Interpretation of the normalized squared Frobenius norm of correlation matrix

Let ${\bf R}$ be a $n\times n$ correlation matrix. I was wondering what is the interpretation of the following norm: \begin{equation} \frac{1}{n}\|{\bf R}\|_F^2 \end{equation} I know that is equal ...
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3answers
215 views

Show that every open subset can be represented as a union of at most countably many balls

Consider $\mathbb{R^n}$ with the $\infty$-norm. Show that every open subset can be represented as a union of at most countably many balls. This might have been answered before but I do not ...
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41 views

How to prove Positivity (property of the Norm in Rn)

Assume v is a vector in Rn. I want to prove that: the magnitude of v >= 0, and magnitude of v = 0 if and only if vector v = 0 vector All I can think of is ...
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5answers
56 views

There are infinitely many vectors such that $\|\mathbf u-(1,1,1)\|\le 3$ and $\|\mathbf u+(1,1,1)\|\le 3$

I want to prove that there are infinitely many solutions in $3$-space for $\|\mathbf u-(1,1,1)\|\le 3$ and $\|\mathbf u+(1,1,1)\|\le 3$ (where bold refers to vectors). My proof: What is wrong with ...
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0answers
22 views

Closed form formula for lower bound on induced max norm of inverse matrix

I am working with the induced max-norm of the inverse of an $n \times n$ matrix $V$ (it is a variance-covariance matrix). Let $||\cdot||_{\infty}$ be the max vector norm. I know: $|| V^{-1} || = ...
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72 views

Relationship between $L^1$ norm and sparsity

I'm doing some research in the field of sparse representation and sparse modeling. I have two variables and their $L^1$ norm is calculated to make comparisons. As I take it the smaller the value of ...
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44 views

The norm of real powers of strictly positive bounded linear operators

Why does one have $\|A^x\|=\|A\|^x$ if $A$ is a positive, linear, bounded operator and $x$ is a real number? By spectral theorem I would deduce $$\|A^x \|=\| {U^*}^x D^x U^x ...
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1answer
57 views

Gradient of the TV norm of an image

Context: I am trying to implement an algorithm for X-ray image reconstruction called ADS-POCS that minimizes the TV norm as well as reconstructs the image. After separating the reconstruction into 2 ...
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44 views

Norm of pointwise product of Lp functions

Does the following inequality hold in $L_p$ spaces? $\|fg\|_p\leq\|f\|_p\|g\|_p$ How would I go about proving this? Do I need to apply Cauchy Schwarz?
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1answer
31 views

Show that the function $A →\|A\|$ defined by $\sup \|Ax\|/\|x\|$ is a norm in the space $M_n$ of $n\times n$ matrices with real entries

Show that the function $A →\|A\|$ defined by $\sup \|Ax\|/\|x\|$ is a norm in the space $M_n$ of $n \times n$ matrices with real entries. Definition 1.26. Let $X$ be a linear space (over $R$). A ...
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1answer
35 views

How is it possible to show that the norm on the elements of Hamilton Quaternions is such that $N(\alpha \beta) = N(\alpha)N(\beta)$?

Let $\alpha,\beta \in \mathbb{H}$ and the norm on $\mathbb{H}$ is defined as $N(\alpha) = \alpha \bar{\alpha}$. How is it possible to show that the norm on the elements of Hamilton Quaternions is ...
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37 views

Simple Euclidean Norm Inequality

I feel rather silly for having to ask this question in specific and am by no means looking for a flat out step by step answer. I understand the definition for the euclidean norm in an n-dimensional ...
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1answer
52 views

How to prove Clarkson's inequality?

I do not know how to prove one of the four Clarkson's inequalities: let $u,v \in L^p(\Omega)$, if $1 < p < 2$, then $$ \bigg\lVert \frac{u+v}{2} \bigg\rVert_p^p + \bigg\lVert \frac{u-v}{2} ...
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1answer
46 views

Are some of the axioms of a norm of a vector space unnecessary?

I have a homework problem where my task is to find out if some of the axioms of a norm of a vector space are unnecessary, meaning they can be derived from other axioms (I presume from the problem ...
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20 views

Why does this “wlog” make sense: $L^p$-norms of random variables

Let $$\overline{X_n}:=\max_{0 \leq m \leq n} X_m^+$$ for a sequence of random variables $X_i, i \geq 1$ (in fact, it is a submartingale), where $X_m^+:=\max(X_m,0)$. We want to show that ...
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1answer
49 views

Proof that a matrix function can be made arbitrarily close to identity

$\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}$ Define the following matrix function: $$ \phi(A) = \sum_{i=0}^\infty \frac{A^i}{(i+1)!} $$ for any $A\in \mathbb{R}^{n \times n}$. A useful ...
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Definition of the sup-norm in the Qing/Fanghua

I am currently studying the book "Elliptic Partial Differential Equations" - Qing / Fanghua, COURANT, in chapter 2, section 2.3. A priori estimates , the autors define by $\alpha$ the sup-norm of ...
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314 views

Cannot understand how angle between two vectors is calculated

On the picture below I am not getting why we calculate $\cos^{-1}(\frac{1}{3})$ instead of $\cos(\frac{1}{3})$. Sorry if the question is dumb.
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Reference for multiplicative norms

Does anyone know of a book which deals with multiplicative norms, e.g. $N( x\circ y)=N( x)N( y)$ ? Any help appreciated. A Google search showed up a few links, but I was hoping for an intermediate ...
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1answer
42 views

Is this measure called anything?

Well, I do not know if this is formally a measure, but imagine $f(\mathbf v_1, \mathbf v_2)$: $$f(\mathbf v_1, \mathbf v_2) = \|\mathbf v_1 - \mathbf v_2\|_{\ell_0}$$ where $\mathbf v_1, \mathbf v_2$ ...
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euclidean norm expressed as a supremum

Reading some article I have come across the following statement: $$ \mathcal{E}\sup_{v \in S} \langle v,g\rangle = \mathcal{E} \|g\| $$ where $g = (g_1,\ldots,g_n)$ for $g_i$ i.i.d. $\mathcal{N}(0,1)$ ...
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11 views

Mahalanobis distance is Euclidian after transformation?

How to prove that Mahalanobis distance for points x and y with covariance matrix S is same as Euclidian distance if the input space would be transform to space where all the axis of covariance matrix ...
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Prove any sublinear $p:X\to\mathbb{R}$ satisfies $p(x)\ge 0$ for all $x\in X$

If $(X,\|\cdot\|)$ is a complex normed vector space, $p:X\to\mathbb{R}$ is sublinear iff $p(\alpha x)=|\alpha|p(x)$ and $p(x+y)\le p(x)+p(y)$ for all $\alpha\in\mathbb{C}$ and $x,y\in X$. I want ...
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1answer
23 views

Whether $||a||_{L^p} +||b||_{L^p}\le||\sqrt{a^2+b^2}||_{L^p}$

$a=a(x), b=b(x)$ are elements of $L^p(\Omega)$, $\Omega$ is bounded open subset of $R^n$. Whether $||a||_{L^p} +||b||_{L^p}\le||\sqrt{a^2+b^2}||_{L^p}$ ?
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Is the Hilbert-Schmidt norm of a matrix induced from a vector norm? [duplicate]

A general way to get a matrix norm is inducing it from a vector norm. The question is, can the Hilbert-Schmidt norm obtained this way? As a norm general question, is every matrix norm obtained this ...
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52 views

Kernel-induced metric

Given a kernel k on input space X defining RKHS (Reproducing kernel Hilbert space) H. Let Φ : X → H denote the corresponding feature map (think of Φ(x) = k(x, .)). Let x, z ∈ X . How can I show that ...
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53 views

How to prove equivalence of two norms?

have linear span $E=Span(1,x,x^2,...,x^{2015})$ on the $C[0,1]$. $$ \left \| f \right \|_{\infty} = \underset{x\in[0,1]}{\max} \left |\sum_{i= 0}^{2015} \alpha_{i} x^{i} \right | $$ $$ \left \| f ...
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Doubts about proof of completeness of finite-dimensional normed spaces

Every $n$-dimensional normed space $X$ (over $\mathbb R$ or $\mathbb C$) is complete. Let $\{e_1,\dots,e_n\}$ be a basis for $X$ and define the norm $\|\cdot\|$ on $X$ by $\|x\|=\max_{i\leqslant ...
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86 views

Two (equivalent ?) norms on Hilbert space

Let $H$ be a vector space equipped with two inner products $\langle \cdot,\cdot\rangle_1, \ \langle \cdot,\cdot\rangle_2$, s.th. $(H,\langle \cdot,\cdot\rangle_j)$ is a Hilbert space for $j=1,2$. ...
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39 views

Lenstra's integer programming algorithm: Finding a lattice point “near the center”

Preliminaries: As part of Lenstra's algorithm for integer programming (see here, page 4) we compute a linear transformation $\tau$ and a point $z \in \mathbb{R}^n$ which meet certain conditions (step ...
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1answer
56 views

Proving that a given finite-dimensional vector space is isometrically isomorphic to $(\mathbb R^n,\|\cdot\|_\infty)$

Let $X$ be an $n$-dimensional vector space with a basis $\{e_1,\dots,e_n\}$. Consider the norm $\|\sum_{i=1}^n \alpha_ie_i\|=\max_{i\leqslant n} |\alpha_i|$ for $x=\sum_{i=1}^n\alpha_ie_i\in X$. We ...
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Inequality of expansion of $\| \ \ \|$

The following is from a proof of a paper Distributed subgradient Methods for Multi-Agent Optimization, Nedic & Ozdaglar - Lemma 5 Note: ...
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1answer
22 views

From “$\text{lim inf}_{k\rightarrow\infty} \|x(k)\|=0 $” to “$\text{lim}_{k\rightarrow\infty} \|x(k)\|=0 $”

Consider the following: If $\ \ \text{lim inf}_{k\rightarrow\infty} \|x(k)\|=0 \ \ $ and $\ \ \text{lim}_{k\rightarrow\infty} \|x(k)\|^2=0\ \ $ then $\ \ \text{lim}_{k\rightarrow\infty} ...
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26 views

proof of $\|y-\sum_{i=1}^n a_ix_i\|\leq \sum_{i=1}^n a_i\|y-x_i\|$

How to prove $\|y-\sum_{i=1}^n a_ix_i\|\leq \sum_{i=1}^n a_i\|y-x_i\|$? $\ \ \ \ \sum_i a_i = 1$, $0\leq a_i \leq 1$ and $y,x_i\in \mathbb{R}^m, \ \ \forall i$ It seems simple however ...
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Does $\|A\|_2\leq \|B\|_2$ imply $\|CA\|_2\leq \|CB\|_2$ or $\|AC\|_2\leq \|BC\|_2$?

Assume matrices $A$, $B$, and $C$ are of same dimensions, does $\|A\|_2\leq \|B\|_2$ imply $\|CA\|_2\leq \|CB\|_2$ or $\|AC\|_2\leq \|BC\|_2$? $\|A\|_2$ denotes by $\lambda_{max}\sqrt{A^TA}$, and ...
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1answer
36 views

Conditional Expectation: Orthogonal Projection in $L^2$

Let $X \in L^2(\Omega, \mathcal{A}, P)$ and $\mathcal{F}\subseteq \mathcal{A}$ a sub-sigma-Algebra. Then $E[X \mid \mathcal{F}]$ is the orthogonal projection of $X$ onto $L^2(\Omega, \mathcal{F}, ...
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22 views

Monotone Convergence Theorem: Application for $p$-norms

Let $X$ be a non-negative random variable in $L^p(\Omega, \mathcal{A},P)$. We have shown that $$||\min(X,M)||_p \leq c$$ for all $M \geq 0$ and want to conclude that $$||X||_p \leq c.$$ In the ...
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1answer
45 views

Operator norm of matrix

Consider $\mathbb{R}^2$ with norm $\Vert (x,y) \Vert=\sqrt{x^2+y^2}.$ I would like to compute the operator norm w.r.t. the above norm of a matrix $$A=\begin{pmatrix} a_{11} & a_{12} \\ a_{21} ...
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1answer
62 views

Prove this integral inequality

Prove this assuming $f$ is integratable: $$\int_{-\pi}^\pi\vert f(t)\vert \, dt\leq \sqrt{2\pi}\sqrt{\int_{-\pi}^\pi\vert f(t)\vert^2}\, dt =2\pi \Vert f\Vert.$$ I tried to square both sides and use ...
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58 views

Why is this vector of unit length?

Given $A$ a symmetric positive definite real matrix and $V$ a non zero column vector define: $B=V^\top A^{-1} V$ and since $A$ is p.d., $B>0$. Also, $A$ has a unique square-root $\sqrt{A}$. I ...
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1answer
23 views

the norm a functional that maps a convergent sequence to its limit

Given the linear space $\vec{E}$ of real convergent sequences, dotated with the norm \begin{equation*} \begin{aligned} \| \cdot \| : \vec{E}&\to \mathbb{R} \\ x=(x_n)_{n\in\mathbb{N}} ...