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

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Norm of integral operator in $L^1$

What is the norm of the operator $$ T\colon L^1[0,1] \to L^1[0,1]: f\mapsto \left(t\mapsto \int_0^t f(s)ds\right) $$ ?
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251 views

p-adic norms and products

I came across the following problems about p-adic norms: Problem. Show that $$\prod_{p} |x|_p = \frac{1}{|x|}$$ where the product is taken over all primes $p = 2,3,5, \dots$ and $x \in ...
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313 views

Cauchy-Schwarz for metrics with arbitrary signatures

When the norm of a vector is always greater than or equal to zero, the Cauchy-Schwarz inequality holds, but what if we look at a metric with an arbitrary signature? Then the inner product of a vector ...
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97 views

A metric on $\mathbb{R}^n$ such that $d(\lambda x, \lambda y)=|\lambda| d(x,y)$ which is not induced by a norm

Let $V=\mathbb{R}^n$. Let $d:V \times V\rightarrow \mathbb{R}$ a metric on $\mathbb{R}^n$. Assume that for any $x,y\in V$ and $\lambda \in \mathbb{R}$, we have $d(\lambda x, \lambda y) = ...
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123 views

a conjecture on norms and convex functions over polytopes

Suppose one has a convex, bounded polytope P $\subset R^n$ and a strictly convex function $f$ defined everywhere on $R^n$. $f$ has a unique minimum; and suppose this minimum occurs somewhere strictly ...
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70 views

Is there a useful relationship between pointwise and $L^2$ distance?

It would be really convenient to get a bound on the point-wise closeness of functions by knowing their $L^2$ distance. Clearly, if two functions are close in the $L^2$ sense, you cannot get a general ...
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62 views

Missing a necessary power in this proof - please help.

This question is somewhat related to Gradient Estimate - Question about Inequality vs. Equality sign in one part. That question was related to part (c) of a problem I am working on, and this question ...
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55 views

Given a matrix $A$ such that $||A||<1$, prove that $I-A$ is invertible

Note: This is a question seen in class while discussing metric spaces and norms, so my recollection might not be 100% accurate. I saw a proof in class, but I wanted to know if there was a different ...
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267 views

Why is the 2 norm “special”? [duplicate]

Out of all the vector norms, the $2$ norm, or the Euclidean norm, seems to be "special". Primarily, I say this because we use the 2 norm as a means of determining the distance from one point to ...
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319 views

Norm in a dual space

If $f \in X^*$, with $X^*$ the dual space consisting of all linear bounded functionals on a linear normed space $X$. With the norm defined as $||f||_{X^{*}} = \sup_{||x|| \leqslant 1} |f(x)|$. Why ...
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382 views

triangle inequality for a certain norm

Let $d$ be a metric on a (say real) vector space $E$, with the property $$d(x,x+cy)=|c|d(x,x+y)$$ for all $x,y\in E$ and scalars $c$. I am trying to prove that $x\mapsto d(x,0)$ defines a norm. The ...
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281 views

Dual to the dual norm is the original norm (?)

I have the following questions about dual norms : How do you prove that the dual of the dual norm is in fact the original norm? This is what I have so far: If I have $\|y\|_* $ as the norm dual of ...
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59 views

Equivalent continuation of a metric

Hello fellow mathematicians, I am confronted with the following, supposedly not too difficult, problem: Let $(E,f_1)$ be a normed space and $F \subset E$ a linear subspace. Let $f_2$ be a norm on E ...
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208 views

Proving an alternative norm on Sobolev space is equivalent to usual norm

I have this exercice and my problel is only in item 4, and i will desespere. Let $f \in L^2(\mathbb{R}^n).$ 1- Why the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admits a unique ...
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406 views

Using Lax Milgram to find a weak solution in an intersection of Sobolev spaces

I am trying to prove the existence of a weak solution of the problem: $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ on the bounded open set $U\subset\mathbb{R}^n$ ...
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411 views

Hilbert Schmidt Norm-Rank-inequality

Problem: Let $A_{n.n}$ be square complex matrix. Prove the following: $$\left \| A \right \|=\left \| A \right \|_{HS}\Leftrightarrow rank(A)\leqslant 1$$. Where $\left \| . \right \|_{HS} $ is the ...
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renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
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133 views

relation between norms of two vectors

Can we say that; if the $l_1$-norm of an arbitrary vector $a$ is smaller that $l_1$-norm of $b$ ($||a||_1 \le ||b||_1$) then the $l_2$-norm of $a$ is smaller than $l_2$-norm of $b$ ($||a||_2 \le ...
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Why does $\|\sum_{i=1}^{N}\langle f,\phi_i\rangle\phi_i\|^2 = \sum_{i=1}^{N}|\langle f,\phi_i\rangle|^2 $?

I'm reading about Fourier analysis and there is one equality, which I don't understand. Why does: $$\left\|\sum_{i=1}^{N}\langle f,\phi_i\rangle\phi_i\right\|^2 = \sum_{i=1}^{N}|\langle ...
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Is norm non-decreasing in each variable?

Let me try again. Suppose $\|\cdot\|$ is a norm in $\mathbb{R}^n$ and let $$f(x_1,...,x_n)=\|(x_1,...,x_n)\|$$ where $x_i\geq 0, \forall i$. I want to prove or disprove that $f$ is an nondecreasing ...
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Norm and square of the ideal $(2,1+\sqrt{-5})$ in the ring of integers

Let $I=(2,1+\sqrt{-5})$ be an ideal of the ring of integers of $\mathbb Q(\sqrt{-5})$. What is its norm $N(I)$? And is $I^2$ principal? My notes say: $1$, $\sqrt{-5}$ is a $\mathbb Z$-basis ...
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On an estimate of sequences with weights

Does there exist a $C > 0$ such that $$ \sum_{n \geq 1} a_n \leq C \left( \sum_{n \geq 1} 2^n a_n^2 \right)^{1/4} \left( \sum_{n \geq 1} 2^{-n} a_n^2 \right)^{1/4} $$ for all $a_n \geq 0$ with ...
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A norm on $\mathbb{R}^2$ such that $\partial C$ is the unit sphere?

Suppose we are on $\mathbb{R}^2$. Assume that $C \subset \mathbb{R}^2$ is a convex bounded neighborhood of the origin invariant by central symmetry. Let $\partial C$ denote the boundary of $C$. My ...
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55 views

Is it true that $2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty}$?

For any $C^1$ function defined in $(0,1)$, is it true that $$ 2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty} $$ If it is true, how to prove it?
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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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100 views

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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143 views

Operator Norm of a Matrix composed of Standard Basis and Fourier Basis

Let $\mathbf{A}_n$ be an $n\times 2n$ matrix (where $n=2^k$) composed of Fourier basis and standard basis; that is, $$\mathbf{A}_n = \begin{bmatrix}\mathbf{I}_n & \mathbf{F}_n\end{bmatrix}$$ ...
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Norm of vector in $\mathbb{R}^3$ with multiple

If we have a vector in $\mathbb{R}^3$ (or any Euclidian space I suppose), say $v = (-3,-6,-9)$, then: May I always "factor" out a constant from a vector, as in this example like $(-3,-6,-9) = ...
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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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369 views

Monotonicity of $\ell_p$ norm

Consider a $n$ dimensional space, it is known (Wikipedia) that for $p>r>0$, we have $$ \|x\|_p\leq\|x\|_r\leq n^{(1/r-1/p)}\|x\|_p. $$ I have two questions about the above inequality. $(\bf ...
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Square matrix $\|Ax-Ay\|\le \|x-y\|$

Could you give me an example of a square matrix $A\in \mathcal{M}_{2 \times 2}$ or $\mathcal{M}_{3 \times 3}$ for which we have $\|Ax-Ay\|\le \|x-y\|$, $ \ \ x, y \in \{0, e_1, . . . , e_n\}, \ \ e_1, ...
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Linear isometry between $c_0$ and $c$

The following question is an exercise and so I'm just looking for advices and not for answers if it's possible. I have the following sets in $l^\infty$ $$c_0 := \{x_n \in l^\infty: \lim x_n = 0\} ...
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Difference between orthogonal and orthonormal matrices

Let $Q$ be an $N \times N$ unitary matrix (it's columns are orthonormal). I understand since Q is unitary, it would preserve the norm of any vector $X$, i.e, $||QX||^2=||X||^2$. My confusion comes ...
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Proof of an inequality of $L^p$ norms

For a general measure space, we define : $\|f\|_p= \left(\int\vert f\vert^p du\right)^{1/p}$. Let $0 < a < b < c < \infty$ and prove the following: $$ \|f\|_b \leqslant \max\{\|f\|_a, ...
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Finding the norm of the operators

How do I find the norm of the following operator i.e. how to find $\lVert T_z\rVert$ and $\lVert l\rVert$? 1) Let $z\in \ell^\infty$ and $T_z\colon \ell^p\to\ell^p$ with $$(T_zx)(n)=z(n)\cdot ...
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$\|f+g\|_p\leq\|f\|_p+\|g\|_p$ where $\|f\|_p=(\int_{a}^{b}|f(t)|^p dt)^\frac{1}{p}$

Using the fact that $$cd\leq\frac{c^p}{p}+\frac{d^q}{q}$$ if $$\frac{1}{p}+\frac{1}{q}=1$$ and letting $$c=\frac{|f(t)|}{\left(\int_{a}^{b}|f(t)|^p dt\right)^\frac{1}{p}}$$ and ...
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47 views

Minkowski type inequality in Banach algebras

Under which circumstances it is true that $\|(A+B)^n\|^{1/n}\le \|A^n\|^{1/n}+\|B^n\|^{1/n}$ for elements $A$ and $B$ in a Banach algebra and a natural number $n$?
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I am not sure how to calculate this norm?

I have the following matrix: $$A= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{bmatrix} $$ What is the norm of $A$? I ...
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Why is minimizing the nuclear norm of a matrix a good surrogate for minimizing the rank?

A method called "Robust PCA" solves the matrix decomposition problem $$L^*, S^* = \arg \min_{L, S} \|L\|_* + \|S\|_1 \quad \text{s.t. } L + S = X$$ as a surrogate for the actual problem $$L^*, S^* ...
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$L_1$ projection of sum of convex functions onto polytopes

Suppose I have a function $f(x) : \mathbb R^n \to \mathbb R$ that is the sum of a given strictly convex function $g : \mathbb R \to \mathbb R$ in a single variable, i.e. $f(x) = g(x_1) + g(x_2) + ...
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What are norms of sub-matrices invariant under a block diagonal similarity transformation of a block matrix?

Say $M := \begin{pmatrix} A & B\\ C & D \end{pmatrix}$ is a block matrix with $A, D$ being square matrices and this $B$ and $C^T$ having the same shape. Is there any norm characterizing the ...
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let $A$ be an n by n matrix, show that $||A||_{OP} \leq ||A||_{HS} \leq \sqrt{n} ||A||_{OP}$

We are given $A \in M_{n}(\mathbb R)$ and the following norms: $||.||_{e}$ is the standard euclidean norm of $\mathbb R^n$. $||A||_{OP}$ is the operator norm of $A$, meaning $||A||_{OP} = ...
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Generalization of $\frac{a + b}{c + d} \leq \text{max}(\frac{a}{c}, \frac{b}{d})$

I'm looking for a matrix version of the basic inequality for the ratio of two sums of positive numbers: $$\frac{a + b}{c + d} \leq \max\left\{\frac{a}{c}, \frac{b}{d}\right\}.$$ Specifically, I have ...
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203 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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norm of inverse less than 1

I just wanna ask if what I am doing here make sense: Let $\epsilon$ be arbitrary positive number. Choosing $\epsilon$ and let it approaches 0, I would like to have $||(I-\epsilon A)^{-1}|| < 1$. ...
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Matrix Norm set #2

As a complement of the question Matrix Norm set and in order to complete the Problem 1.4-5 from the book: Numerical Linear Algebra and Optimisaton by Ciarlet. I have this additional conditions: (3) ...
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382 views

Poincaré's lemma with norm in $H_{0}^{1}$

I wonder why $\| u\|_{H_0^1} = \int_{\Omega} |Du|^2$ for u in $H_0^1(\Omega)$, with $\Omega = (-1,1)$? I might be wrong but isn't $\| u\|_{H^1} = \| u\|_{L^2} + \| Du\|_{L^2}$? How come that $\| ...
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297 views

Prove a basic fact on a linear combination of vectors

Let $x_i (i=1,...,n, n>d)$ be a unit vector in $R^d$. $c_i>0$ is a positive real scalar. How to prove the following fact? Fact: There exist some vectors $x_i$ such that $\sum_{i=1}^n c_i ...
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norm of differential operator on $P^n[0,1]$

Consider the space $P^n[0,1]$ of polynomials of degree $\leq n$ on $[0,1]$, equipped with the sup norm. Now, this is a finite dimensional space, so all linear operators have to be continuous, hence ...
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135 views

Effect of doubly stochastic matrix on vector norm

Let $D$ be a $N \times N$ doubly stochastic matrix, $x$ be a $N$ dimensional vector. What is the relation between $\Vert Dx \Vert_2$ and $\Vert x \Vert_2$? In addition if $\Vert x \Vert_2=1$, what ...