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

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Monotone matrix norms

[Ciarlet 2.2-10] Let $\mathscr{S}_n$ be the set of symmetric matrices and $\mathscr{S}_n^+$ the subset of non-negative definite symmetric matrices. A matrix norm $\|\cdot\|$ to be monotone if ...
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106 views

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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87 views

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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101 views

What are norms used for?

These two questions are quite similar to this one, so I apologise if this irritates anyone. Also, I suspect that a lot can be said in the answer, so I am really just looking for some main points ...
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40 views

Norms on $\mathbb{Q}$

So with respect to the metric $d(x,y)=|x-y|$ induced by the standard absolute value, the real numbers can be constructed as a completion of $\mathbb{Q}$. With respect to the metric $d_p(x,y)=|x-y|_p$ ...
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134 views

Two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on $X$, and a function $f:X\to Y$ Fréchet differentiable with one of the norms but not with the other one?

There is a theorem that if $f : (X,\|\cdot\|_{X1}) \to (Y,\|\cdot\|_{Y1}) $ is Fr├ęchet differentiable, then replacing the norms with some equivalent norms $\|\cdot\|_{X2}$ and $\|\cdot\|_{Y2}$ ...
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97 views

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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194 views

Prove this equality in a Hilbert Space $H$ with Riesz's Representation Theorem.

For $x \in H$, prove that $\sup_{\|z\| = 1} \langle x , z \rangle= \| x \|$ Here is a quick one. If someone could improve this it would be great Proof By Cauchy Schwarz, $\langle x,z \rangle ...
3
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164 views

The norm of an operator

Let $\rho(x)$ be a weight function in a unit sphere, such that \begin{equation} \begin{array}{l} \displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\ \displaystyle 2. \rho(x)\in ...
3
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73 views

Exercise from textbook about norm

The below diagram included a relevant page and the question 4 of the exercise which i am not sure how to do. Any hint or solution are welcome Attempts: a) I have done it b) i have tried to show that ...
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91 views

Is $\rho(f) := \mu(|f| \geq \epsilon), \epsilon >0$ some kind of generalized norm?

Let $\rho(f) := \mu(|f| \geq \epsilon)$ where $\epsilon >0$ is a constant and $\mu$ is a measure and $f$ is a measurable function. Then $f\equiv 0 \rightarrow \rho(f) = 0$ when $|a| \neq 0$, ...
3
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99 views

Confusion about matrix norms

Reading the wiki article I get confused about matrix norms. My question, is it true that $$\lVert Mx \rVert \leq C(\sup_{ij}{M_{ij}}) \lVert x \rVert$$ where $M$ is a matrix and $x$ is a vector and ...
3
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222 views

Minimizing maximum absolute column sum norm of the residual between a matrix and its $k$-rank approximation

Let $X \in \mathbb{R}^{m\times n}$ be a matrix with rank $r$. How can we find the optimal $\tilde{X} \in \mathbb{R}^{m\times n}$ whose rank is $k$ where $k\leq r$ and the reconstruction error in ...
2
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22 views

Bounding the norm of Gaussian random matrix

Suppose $A\in\mathbb R^{n\times m}$ is a random matrix with $n < m$, and each entry $A_{ij}$ follows i.i.d. Gaussian distribution $N(0,1/n)$. I want to know whether we can upper bound the spectral ...
2
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28 views

Is this dual-spaced norm based on $L_2$ norm

I am reading the book of Claes Johnson about Numerical Solution of Partial Differential Equations by the Finite Element Method and particularly pages 34 and 98. I wrote these notes to my craft Is ...
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130 views

Using the Sobolev-Nirenberg-Gagliardo inequality in a proof

If $1 \leq p < n$. The Gagliardo-Nirenberg-Sobolev inequality states that there exists a constant C such that $||u||_{L^{p^{*}}(\mathbb{R}^{n})} \leq C||Du||_{L^{p}(\mathbb{R}^{n})}$ for all $u ...
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60 views

Asymptotic behavior of $L^2$ norm for increased matrix dimensions

I am playing with matrices which are linear combinations of identity matrix, Pauli spin matrices, $\sigma_x$ and $\sigma_z$ or their tensor products. For example, let the matrix be $H$. So, $H$ could ...
2
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102 views

Operators from $L^{\infty}$ to $L^{\infty}$, below bound of the norm

If $T(f)(x)=\int K(x,y)f(y)dy$, where $K(x,y)$ is locally integrable, is bounded on $L^{\infty}$, how can we show that $\|\int|K(\cdot,y)|dy\|_{L^{\infty}}\le \|T\|_{L^{\infty}\rightarrow ...
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55 views

Find a matrix that minimizes the norm of a linear function

Let $S=AA^{T}+XX^{T}-AB^{T}\left(BB^{T}\right)^{-1}BA^{T}$ where $A$ and $B$ are known. What is the best possible $X$ that minimizes $\left\Vert S^{-1}y\right\Vert _{2}$ for any vector $y\in ...
2
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59 views

Proving norm inequality with Schwarz's inequality

I'm stuck on the following problem: Suppose that $\{V_j : j \in \mathbb{Z}\}$ is a multiresolution analysis with scaling function $\phi$, and that $\phi$ is continuous and compactly supported. ...
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74 views

Are these two definitions for dual norm equivalent

Suppose there is a p norm, also is a function $\left\|x\right\|_p$, the dual function at $z$ of it is, $$\sup\limits_x\left\{ z^{T} x-\left\|x\right\|_p\right\}$$ The second is, $$ \sup\limits_x ...
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43 views

Finding an orthornormal basis given a bilinear form

Take $ \langle \cdot,\cdot\rangle$ to be a nondegenerate symmetric bilinear form (which may not be positive definite). Take $\Vert x\Vert = \vert \langle x,x\rangle \vert^{1/2}$. I want to show that ...
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80 views

Examples of functions that are Lipschitz w.r.t. Schatten p-norm?

A convex function $f$ is $R$-Lipschitz w.r.t. to a norm $\|\cdot\|$ if for all points $a, b$ we have $|f(a)-f(b)| \leq R\|a-b\|$. For a real symmetric $n\times n$ matrix $A$ with eigenvalues denoted ...
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155 views

Step functions dense in Integrable functions with respect to $L_2$

Let $I$ be a bounded interval. Prove that $\{\text{step functions }I \to C\}$ is dense in $\{\text{integrable functions }I \to C\}$ (Riemann Integrable) with respect to $\|.\|_2$ ($L_2$ norm)
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613 views

Equivalence of norms is a equivalence relation

Two norms $||-||_1 $, $||-||_2$are equivalent if: for two constants $a,b$ and $x$ from $V$ a vector space over a field it holds that : $$a||x||_1\leqslant ||x||_2\leqslant b||x||_1.$$ This is a ...
2
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315 views

Bound on inverse operator

Define $X = {C^{2, \alpha}}(U \times [0,T])$ and $Y = {C^{0, \alpha}}(U \times [0,T])$ where $U$ is some real interval. Let $F:X \to Y$ be a map. Let $DF(g):X \to Y$ be a bounded linear operator for ...
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108 views

Quick question: finite extensions and norms

[Edit: "Suppose K is a field complete with respect to a discrete valuation, with valuation ring $\mathcal{O}$."] I'm trying to solve the following problem: let $L/K$ be a finite extension of fields, ...
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35 views

An Orthogonal Projection with Weighted Norm

In the context of solving a convex program via projected gradient descent i am facing the following problem: $$\min_{x\in\mathbb R^2}\lVert x-y\rVert_M^2,\qquad\lVert x\rVert\le1$$ or written ...
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77 views

How can I define a measure of similarity between two line segments in $\mathbb{R}^2$?

I have two segments in $\mathbb{R}^2$ and I would like to define a measure of similarity between the two segments. My idea is that I can apply: a scale transformation $s$ in order to equate the ...
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45 views

Find norm of linear operators

I have to check if those operators are bounded and if so what are their norms. 1) $\phi:C^1[0,1]\ni f > \rightarrow\int_0^{1/2}f(t)dt+f'(2/3)\in\mathbb{R}$ with norm ...
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23 views

Show that a function $\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty})$, $f \rightarrow f'$ is continuous

I have the following: Show that the function $$\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty}),~~f \rightarrow f'$$ is continuous. With $$\|\cdot\|_\infty = \sup\{|f(x)| ~ \big| ~x\in ...
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71 views

Showing 2- norm of Ax,squared is equal to $$ x^TM^TMx $$

So, I am trying to prove $$\|Mx\|^2 =x^TM^TMx$$, however I am running into some difficulties. Here, $M \in \mathbb{R}^{m \times n}$ and $x \in \mathbb{R}^n$. I know that when you take the transpose ...
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89 views

Operator norm of the inverse

If I made no mistake, one can calculate the operator norm of the inverse of any given (invertible) operator $A: V\rightarrow V$ via: \begin{align}\|A^{-1}\| & = ...
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49 views

proving uniform convergence for a uniformly continuous function

I've been stuck on this question (2a) for about a day now! Question and havn't really gotten anywhere, I think that the series is that integral of a step function that approximates f ie: $\Gamma_n ...
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70 views

Norm inequality with wedge product

Anyone could help me to prove this following inequality? $\displaystyle\frac{||(u+v)\wedge w||}{||u+v||}\le \frac{||u\wedge w||}{||u||} +\frac{||v\wedge w||}{||v||} $ where $u\wedge v$ is the wedge ...
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29 views

Uniformly Bounded in a Schauder Norm

Suppose $\{h_{k}\}$ is a sequence of elements in $C^{2,\alpha}$. I want to show that $\{h_{k}\}$ is uniformly bounded in $C^{2,\alpha}$. Now, if I'm not mistaken, then $$||h_{k}||_{\displaystyle ...
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29 views

Topological equivalence of any norm on $\mathbb C^n$

In University I have been told that every norm on $\mathbb C^n$, for any $n\in\mathbb{N}$, is equivalent to every other such norm. I have a proof for this on any vector space on $\mathbb R$. Trouble ...
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33 views

Equivalence of norm in a ring.

For the ring $$A=\mathbb{Z}[i\sqrt{3}]=\{a+i\sqrt{3}b:a,b\in \mathbb{Z}\}$$ I had to show that the only invertible elements are $1$ and $-1$, using the norm $$N:\mathbb{Z}[i\sqrt{3}]\quad ...
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78 views

Natural matrix norm of an inverse matrix

Let $\left\|\cdot\right\| : \text{GL}(n,\mathbb{R})\to\mathbb{R}_{\ge 0}$ denote the natural matrix norm, i.e. $$\left\|A\right\|:=\max_{x\ne ...
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109 views

About $f(s)=\sum_{a^2+b^2>0} \frac{(-1)^{a^2+b^2}}{(a^2+b^2)^s}=0$ and the Extended Riemann Hypothesis.

Let $s$ be a complex number with a strictly positive real part ($Re(s)>0$). Let $f(s)=\sum_{a^2+b^2>0} \dfrac{(-1)^{a^2+b^2}}{(a^2+b^2)^s}$ where the sum runs over all positive integers $a,b$ ...
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37 views

Solving a linear matrix equation with respect to the maximum of the euclidian distances between rows.

With $n>m$, real number matrices $A$, $B$, $C$ are shaped like: $$A=\left( \begin{array}{ccc} a_{1,1} & \cdots & a_{1,m} \\ \vdots & \ddots & \vdots \\ a_{n,m} & \cdots ...
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53 views

What is the correct notation for defining norms in measure spaces?

For a function $f \in L_2(R)$, we can define its norm as $$ \|f\|_2^2 = \int f^2(x) dx $$ If I use a different measure $\mu$, I can in turn define the norm as $$ \|f\|_2^2 = \int f^2(x) \mu (x) dx ...
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35 views

Operators and Differential Equations

I have a question about m-dissipative operators. Thus let $T$ be an closed operator with dense domain. Then $(Tx,x)\leq0$ for all $x\in D(T)$ and $\lambda I-T$ surjective for all $\lambda>0$. I ...
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166 views

Minimum L1 norm may not obtain the sparsest solution?

Here is a paper called For Most Large Underdetermined Systems of Equations, the Minimal L1-norm Near-Solution Approximates the Sparsest Near-Solution. However, I did not quite get its definition of ...
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56 views

How to solve an optimization problem with non-convex Frobenius norm constraint?

The form of my problem is: $$ \min_W \|Y-WX\|_F^2-\|V-WU\|_F^2 $$ $$ s.t. \|W\|_F=1 $$ All five variables are matrices. Since the norm constraint is a non-convex one, I have no idea how to solve this ...
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45 views

inequality about Inner product and norm

If $m\times n~(m<n)$ matrix $A$ satisfy the following condition $(1-\delta)||s||_2^2\leqslant \|As\|^2_2\leqslant (1+\delta)\|s\|_2^2$ for all the $n \times 1$ vector with no more than $k$ nonzero ...
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95 views

Triangle inequality for $\sqrt{x^T\cdot A\cdot x}$

I want to prove that $f(x):= \sqrt{x^TAx}$ is a norm, $A \in \mathbb{R}^{n \times n}$ positive definite, $x \in \mathbb{R}^n$. I already proved its positivity and absolute homogeneity, but I don't ...
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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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21 views

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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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 ...