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

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Vector norms on $\Bbb R^n$

Let $u=(u_1,u_2,\ldots,u_n)$ and $v=(v_1,v_2,\ldots,v_n)$ be two vectors in $\Bbb R^n$. Suppose $\left|v_i\right|>|u_i|$ for all $i$. Let $\| \cdot\|$ be any vector norm on $\Bbb R^n$. Is it true ...
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evaluating norm of sum of roots of unity

let $l_1,...,l_n$ be roots of unity. I want to prove that the norm(the product of all conjugates)of $a=l_1+...+l_n$ is not greater than $n$, not smaller than $-n$. how can I do to prove this?
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Is every normed vector space, an inner product space

Let $V$ be a vector space over $\mathbb{C}$. If $V$ is an inner product space, then $V$ is normed (where the norm is defined as $\|x\|=\sqrt{(x,x)}\,\,$). Now if $V$ is normed, does it follow that ...
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Equivalency two norms?

Suppose that the following norms on $C^1[0,1]$ . Are they equivalent norms? $\|f\|=\|f\|_\infty+\|f'\|_\infty$ and $\|f\|=\max\{\|f\|_\infty, \|f'\|_\infty\}$ such that $f\in C^1[0,1]$ , ...
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In $C(\mathbb{R})$, the vector space over $\mathbb{R}$ of continuous functions from $\mathbb{R}$ to $\mathbb{R}$

In $C(\mathbb{R})$, the vector space over $\mathbb{R}$ of continuous functions from $\mathbb{R}$ to $\mathbb{R}$, let $W$ = { f ∈ $C(\mathbb{R})$ | $∫_{-∞}^∞$ | f(x) | $dx$ < ∞} where the ...
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65 views

bounding operator norm by quadratic forms on orthonormal basis

Let $\{u_i\}_{i=1}^n$ be a set of orthonormal basis and $M$ a symmetric matrix. Suppose \begin{equation} \left|\langle u_iu_i^T,M\rangle\right|\leq \tau\ \ \forall i=1,\dots,n. \end{equation} Can I ...
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Norms that $C([0,1])$ to be an incomplete normed space.

I searched all of norms that $C([0,1])$ to be incomplete normed space. But I found only $\|.\|_p$ (for every $1\leq p<\infty$). Are you know another norm on $C([0,1])$ that $C([0,1])$ to be ...
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How to prove $||A||_2\leq ||A||_F \leq \sqrt{n}||A||_2$ [duplicate]

$A$ is a square matrix with dimension $n$ and $||A||_F$ is Frobenius norm.
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Additive norm $||a+b||=||a||+||b||$

I've read somewhere that there exist spaces where $||a+b||=||a||+||b||$ is true iff $a = \lambda b, \ \ \lambda>0$. Could you tell me what spaces have that property and what spaces don't? ...
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Why is the matrix norm $||A||_1$ maximum absolute column sum of the matrix.

By definition, we have: $||V||_p=\sqrt[p]{\displaystyle \sum_{i=1}^{n}|v_i|^p}$ $||A||_p=sup\frac{||Ax||_p}{||x||_p}$, $x\not=0$ and if $A$ is finite, we change sup to max. However I don't really ...
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38 views

How to solve linear system of equations in 1 and inf-norm?

I have the problem to find a linear program that is equivalent to solving the problem that finds a minimum for $||Ax-b||_1$ and $||Ax-b||_{\infty}$. We defined a linear program as follows: $min_{x} ...
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61 views

Operator attains norm

We have the following linear and well-defined mapping $T_{a}(b):=\sum_{j=1}^{\infty}a_{j}b_{j}$, with $\{a_{j}\}\in \ell^{1}$ and $\{b_{j}\} \in c_{0}$. Show that $\|T_{a}\|=\|a\|_{1}$. My work: ...
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46 views

Infinite number of induced norms

While proving that a norm had to come necessarily from a scalar product I have started to wonder about the concept and uniqueness of induced norm. My teacher hasn't clarified to me this doubt, saying ...
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229 views

Bound on the norm of a matrix exponential in Jordan Form

I'm looking to prove the following lemma: Let $A$ be a matrix in $\mathbb{R}^{n\times n}$. Then for any $\lambda^* > \max_{\lambda} \; \mathrm{Re} \; (\lambda)$ such that $ \lambda \in\sigma (A)$, ...
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50 views

Maximum coordinate of a linear transformation of a vector

Given a vector $x \in R^n$ (variable) and a constant matrix $M \in \{0, 1\}^{m \times n}$ (known). $M$ is a binary matrix, meaning that its entries are either $0$ or $1$. I need to obtain an ...
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113 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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78 views

Trouble in proving that $\|x\|_p = \max|x_j|$

We define p-norm in this way: $\|x\|_p = \{\sum ^N_j_=_1|x_j|^p\}^ {1\over p}$ We know that It change to $\|x\|_p = \max|x_j| $ when $ p \to \infty $ How can I prove this ?
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115 views

Finding matrix norm equivalence constants

I've been given the following: "Find the best positive constants $\alpha$ and $\beta$ such that $\alpha\left\|A\right\|_2\leq\left\|A\right\|_1\leq\beta\left\|A\right\|_2$ for all ...
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167 views

Closed-form expressions for dual norms of real normed vector spaces

Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by $\|v\|$. Then say that $V^*$ is the dual space of linear functionals on $V$. The "dual ...
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When does $\|z^2\|=\|z\|^2$

Let $k \in \mathbb{Z}$ and consider the field extension $K := \mathbb{Q}[\sqrt{k}]$. Define a norm on $K$ given by $\|p+q\sqrt{k}\| := \sqrt{p^2+q^2}$. For any $z \in K$, I was interested to know when ...
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81 views

Simple question about matrices

My question is simple : If one replaces some of the entries of a matrix by 0, does he obtain necessarily a matrix with a lower norm? I have to precise that the norm I use is the maximum of the ...
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Compute $\|A(t)\|$

When I do my homework (stability theory), I must use the knowledge to the norm of matrix. But I don't remember it (I mean that I'm not sure). My problem: For matrix $A(t)$ is continuous. Compute ...
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Show that $D =\{ x + y \mid x \in (0,1) ,y \in [1,2) \}$ is open or closed

I have $\mathbb{R}$ with the euclidian metric $|x-y|$ for $x,y\in \mathbb{R}$. I want to show with arguments that the set $D =\{ x + y \mid x \in (0,1) ,y \in [1,2) \}$ is open or closed. As a ...
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Proving norm equivalence in $W^{1-1/p,p}(\Omega)$

Define for $p\in [1,\infty)$ and $\Omega=(0,1)^N\subset\mathbb{R}^N$ $$W^{1-1/p,p}(\Omega)=\left\{u\in L^p(\Omega): \ \int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy<\infty\right\}$$ ...
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50 views

Differential Operators over the space of Analytic Functions

Let $\mathcal{A}(-a,a)$ be the vector space of functions that are analytic on the interval $(-a,a)$ Is there a common topology to place on this space, if yes what is the topology and is it induced ...
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202 views

How to determine a operator norm?

How to solve following: In Hilbert space $W_2^1=\{f:[0,1]\rightarrow \mathbb{C}|f\in AC[0,1], f'\in L^2[0,1]\}$ with scalar product $(f,g)=\int_0^1 f\overline{g}dx+\int_0^1 f'\overline{g'}dx$ is ...
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72 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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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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72 views

Difference: normed space and normed linear space.

I'm currently reading up on Banach spaces. My book "Introduction to Banach spaces and their Geometry" by Beauzamy mentions "normed spaces" in some places, and "normed linear spaces" in other. I really ...
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85 views

How does showing that all norms are strongly equivalent imply that the identity map and the norm map are continuous functions?

In the begging of my introduction to functional analysis course the lecture started with a proof that all norms on $\mathbb{R}^{n}$ are strongly equivalent. Then the lecture said that from this we ...
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1answer
98 views

A basic doubt on the definition of induced matrix norm

In an optimization book I am following, the induced norm is defined as the maximum of the norms of the vectors $Ax$ where the vector $x$ runs over the set of all vectors with unit norm. Now, it says ...
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Counterexample using counting measure

While proving that the norm of the mulplicative operator from $L^2(X) \to L^2(X)$ is the essential supremum of $|g|$ where $g \in L^\infty(X)$, I found that I need the $\sigma$-finiteness of the ...
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45 views

Equality with norms of matrices

I have a problem with prooving of following equality: $$\|E(I-\frac{ss^T}{s^Ts})\|_F^2=\|E\|_F^2-\frac{\|Es\|^2_2}{s^Ts},$$ where $E\in\mathbb{R}^{n\times n}$ and $0\neq s\in\mathbb{R}^n$. I tried to ...
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67 views

Norm of convolution of $n$ Gaussians

If $$f(x)=e^{-(\pi x)^2}$$ and $$\psi_n(x)=(f* f*\dots*f)(x)$$ ($n$ times convolution). Show that $$\lVert \psi_n(x)\rVert = 1$$ (norm in $L^1(\mathbb{R})$). I've tried using the Fourier ...
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1k views

Proof of matrix norm property: submultiplicativity

I've been searching for the definition of the submultiplicative (I think it has multiple names from what I've seen) property in proof form. Some books define it as part of the properties that define ...
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Matrix Norms Inequality Proof

How do I prove this inequality / equivalence about matrix p-norms? It appears on the wikipedia and mathworld.wolfram pages on matrix norms without proof. $\|A\|^2_2 \leq \|A\|_1 \|A|\|_\infty$ Maybe ...
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1answer
78 views

Prove norm inequality: $\|\mathrm x\|_2 \le \|\mathrm x\|_1$

On $\Bbb R^n$, define for $\mathrm x = (x_1, x_2, \ldots , x_n)$ a norm $$\|\mathrm x\|_1 := |x_1| + |x_2| + \cdots + |x_n|$$ By denoting the usual norm by $\|\mathrm x\|_2$, show that $\|\mathrm ...
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1answer
183 views

Matrix Norm Bounds

A natural consideration of matrix norms is to compare them, and one of the many standard results on the induced 1, 2, and $\infty$-norms indicate that $$\frac{1}{\sqrt{n}}\|A\|_\infty\leq \|A\|_2\leq ...
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Help showing $\phi _k$ is a bounded linear functional

Let $V$ be the space of continuous functions on the interval $[-\pi , \pi]$ with the $L^2$ norm $$\lVert f\rVert_2=\left(\int_{-\pi}^\pi |f(t)|^2\mathrm dt)\right)^\frac{1}{2}$$ For $f$ in $V$, define ...
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Prove that two norms are equivalents

Two norms $\|\bullet\|_1$ and $\|\bullet\|_2$ are equivalents iff $\;\exists\;c_1,c_2>0$ such that $c_1\|x\|_1\le \|x\|_2\le c_2\|x\|_1$ We're working in $\mathcal C^1[0,1]$, and I have ...
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1answer
30 views

Norm of difference of two squares of matrices

Let $x,y$ be square matrices and $c$ be any scalar. Is it true that $ \Vert x^2 \Vert - c^2 \Vert y^2 \Vert = \Vert x - cy \Vert ^2$? If this is true then I'm done with the proof of a theorem on ...
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65 views

A question about a proof for why $\|x\|:=\inf\{\lambda>0\mid\frac{x}{\lambda}\in B\}$ is a norm

I started studying functional analysis, a claim that was thought is the second lecture claims that: Let $X$ be a vector space, $B\subseteq X$ is convex, symmetric around $0$ and s.t ...
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393 views

If two norms are equivalent on a dense subspace of a normed space, are they equivalent?

Given a vector space $V$ equipped with two norms $|\cdot|$ and $||\cdot||$ which are equivalent on a subspace $W$ which is $||\cdot||$-dense in $V$, are the two norms necessarily equivalent? The ...
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3answers
572 views

2-norm vs operator norm

I have read that we define the "2-norm" of a matrix as $$\max_i \,{|\sigma_i|},$$ which I have also heard called the "operator norm" (here $\sigma_i$ are the singular values). Also we have the ...
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172 views

A counter example of best approximation

Construct a point $f\in C[0,1]$ and a closed subspace $V\subset C[0,1]$ such that $f$ does not have a best approximation in $V$. Definition: $C[0,1]$ is the set of countinous function with the norm ...
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What are some uses for other norms on $\mathbb{R}^n$

We all know and love the standard $1,2,$ and $\infty$-norms on $\mathbb{R}^n$. However, I have never seen anyone mention uses for any of the other $k$-norms that I'm defining as ...
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39 views

Norm of a mapping

$$C[0,1]=\{f:[0,1]\rightarrow R | \text{$f$ is continuous function}\}$$ $$\|f\|=\max \{|f(x)|:x\in [0,1]\}$$ $$A:(C[0,1],\|\|)\rightarrow(C[0,1],\|\|)$$ $$A(f)(x)=(x^4-x^2)f(x)$$ I have to ...
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derivation of vector norm

what would be the differentiation of this equation :- $F(A) = \sum_{i} \left \| Y_{i} - AB_{i} \right \|^{2} + \lambda \left \| A - C \right \|^{2}$ wrt to A . Y is a column vector and B is column ...
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68 views

Grid Function Norm

I am studying about the global truncation error in finite difference methods and I have a question about calculating the error in a Boundary Value Problem (BVP). If we take a simple 1-D problem, the ...
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1answer
73 views

How an $\ell_1$ Inequality Implies Equality

Suppose that for scalar $\epsilon$ we know that $\vert \epsilon \vert$ is small enough such that the sign pattern on $\mathbf{x}\in\mathbb{R}^n$ is equal to that on $\mathbf{x} + \epsilon \mathbf{h}$, ...