Norm Definition and 263 Threads

Suppose I have a function f(x) \in C_0^\infty(\mathbb R), the real-valued, infinitely differentiable functions with compact support. Here are a few questions: (1) The function f is trivially uniformly continuous on its support, but is it necessarily uniformly continuous on \mathbb R? (2) I...

Hi, With the following norm inequality: ||Av|| ≤ ||A||||v|| implies ||A|| = supv [ ||Av||/||v|| ] I understand that sup is the upper bound of a set B, or least upper bound if B is a subset of A, where the upper bounds are elements of both B and A. Is this saying that the norm of A...

Homework Statement Find two vectors in R4 of norm 1 that are orthogonal to the vectors u = (2, 1, −4, 0), v = (−1, −1, 2, 2) and w = (3, 2, 5, 4). Homework Equations The Attempt at a Solution What i did was, i let a vector x = (x1, x2, x3, x4) that has a norm of 1 and...

Hi everyone, :) Here's a question with my answer, but I just want to confirm whether this is correct. The answer seems so obvious that I just thought that maybe this is not what the question asks for. Anyway, hope you can give some ideas on this one. Problem: Let \(X\) be a finite...

Homework Statement . Let ##X=\{f \in C[0,1]: f(1)=0\}## with the ##\|x\|_{\infty}## norm. Let ##\phi \in X## and let ##T_{\phi}:X \to X## given by ##T_{\phi}f(x)=f(x)\phi(x)##. Prove that ##T## is a linear continuous operator and calculate its norm. The attempt at a...

Real Analysis, L∞(E) Norm as the limit of a sequence. || f ||_{\infty} is the lesser real number M such that | \{ x \in E / |f(x)| > M \} | = 0 ( | \cdot | used with sets is the Lebesgue measure). Definition: For every 1 \leq p < \infty and for every E such that 0 < | E | < \infty we...

Homework Statement This isn't actually a homework question but i thought this would be the right place for it... I am doing exam review and this question is giving me difficulties. Consider the 3x3 diagonal[1,3,1] matrix A. Find nonzero vectors x in ℝ^{3} such that ||Ax||_{3} = ||A||_{3}...

Homework Statement Let \textbf{A} be an m x n matrix and \lambda = \max\{ |a_{ij}| : 1 \leq i \leq m, 1 \leq j \leq n \}. Show that the norm of the matrix ||\textbf{A}|| \leq \lambda \sqrt{mn}. Homework Equations The definition I have of the norm is that ||\textbf{A}|| is the smallest...

Homework Statement Prove ∥A∥F =√trace(ATA), for all A ∈ R m×n Where T= transpose Homework Equations The Attempt at a Solution I tried and i just can prove it by using numerical method. Is there anyway to prove the equation in a correct way?

Why is it that the norm of a vector is written as a "double" absolute value sign instead of a single one? I.e. why is the norm written as $ || \vec{v} || $ and not $ | \vec{v} | $? I think $ | \vec{v} | $ is appropriate enough, why such emphasis on $ || \vec{v} || $? I think it's rather natural...

Homework Statement 3 randomly selected observations form the standard normal distribution are selected. What is the probability that their sum is less than 2? Homework Equations The Attempt at a Solution I know that the answer is 0.874928, but I don't know how to get that...

I am struggling with this question. I need a different perspective. Any recommendation is appreciated. Please click on the attached Thumbnail.

Homework Statement So this is part of a couple of questions. Find the exact length of p, of OP, by considering a dot product with OP.(Hint OP will be orthogonal to the plane.) Hence find the position vector of P ( P is is the point on the plane closest to the origin F.Y.I) Homework...

Hi, I don't get which of the many matrix norms is invariant through a change of basis. I get that the Frobenius norm is, because it can be expressed as a function of the eigenvalues only. Are there others of such kind of invariant norms? Thanks

Greetings everyone! I have a set of tasks I need to solve using using operator norms, inner product... and have some problems with the task in the attachment. I would really appreciate your help and advice. This is what I have been thinking about so far: I have to calculate a non trivial upper...

If I have the following operator for $H=L^2(0,1)$:$$Tf(s)=\int_0^1 (5s^2t^2+2)(f(t))dt$$ and I wish to calculate $||T||$, how do I go about doing this: I know that in $L^2(0,1)$ we have that relation:$$||T||\leq \left ( \int_0^1\int_0^1 |(5s^2t^2+2)|^2dtds\right )...

Hi all, I have a basic optimisation question. I keep reading that L2 norm is easier to optimise than L1 norm. I can see why L2 norm is easy as it will have a closed form solution as it has a derivative everywhere. For the L1 norm, there is derivatiev everywhere except 0, right? Why is this...

Hello, I've often read that average BMR can decrease as a result of a restrictive diet, in a process known as adaptive thermogenesis. What I haven't been able to find out though, is how exactly the body manages to 'use fewer nutrients' for the same tasks. Are sacrifices made anywhere? Are...

To find the closest point to b in the space spanned by the columns of A we have: \mathbb{\hat{x}}=(A^TA)^{-1}A^T\mathbb{b} My question is, shouldn't this solution ##\hat{x}## depend on the choice of distance function over the vector space? Choosing two different distance functions might give...

Homework Statement u=(2,-2,3) v=(1,-3,4) w=(3,6,-4) 1. \left \| 2u-4v+w \right \| 2. \left \| u \right \|-\left \| v \right \| The Attempt at a Solution 1. \left \| 2(2,-2,3)-4(1,-3,4)+(3,6,-4) \right \| \left \| (4,-4,6)+(-4,12,-16)+(3,6,-4) \right \| \left \|...

Homework Statement Let H= C[-1,1] with L^2 norm and consider G={f belongs to H| f(1) = 0}. Show that G is a closed subspace of H. Homework Equations L^2 inner product: <f,g>\to \int_{-1}^{1}f(t)\overline{g(t)} dt The Attempt at a Solution I've been trying to prove this for a...

Hello, I am trying to write a mtlab code to compute Frobenius norm of an mxn matrix A. defined by ||A||_{F} = \sqrt{ \sum_{i=1}^m \sum_{j=1}^n a^{2}_{i,j}} I have so far written this code, but it does not work, if anyone can help /guide me to the right path, would be greatly...

I quote an unsolved question from MHF posted by user Civy71 on February 18th, 2013

Are the axioms of a Norm different from those of a Metric? For instance Wikipedia says: a NORM is a function p: V → R s.t. V is a Vector Space, with the following properties: For all a ∈ F and all u, v ∈ V, p(av) = |a| p(v), (positive homogeneity or positive scalability). p(u + v) ≤ p(u)...

Homework Statement I want to show that for an n x n matrix A with complex entries, if \left\|Ax\right\|=\left\|x\right\| for any vector x in C^n, then the rows of A are an orthonormal basis of C^n. Homework Equations The Attempt at a Solution All I've managed to do so far is show...

So I'm taking some courses in calculus, and I am surprised by how little explaining there is to the definition of the euclidean norm. I have never understood why you want to define the length of a vector through the pythagorean way. I mean sure, it does seem that nature likes that measure of...

Prove that the function $f : \mathbb{R}^2→\mathbb{R}$ defined by $f(x)=\left\{\begin{matrix} \frac{|x|_2}{|x|_1} , if x\neq 0 \\ a, if x = 0\end{matrix}\right.$is continuous on $\mathbb{R}^2$\{$0$} and there is no value of $a$ that makes $f$ continuous at $x = 0$.

Here is the question: Here is a link to the question: Proof about norm and continuous function. Help!~~~~~~~Hurry!~~~~? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Let's say I have a vector (4+2i, 1-i), how do I take an L2 norm? Dont tell me I simply do sqrt(16+4+1+1)..?

My book goes on to say: "If we consider both C^n and C^m with norms, then we define the norm of an M x N matrix A by.." Then the formula says norm of A=sup (over abs(v)=1) of abs(Av) = sup (over v does not equal 0) abs(Av)/abs(v) Can someone please provide me at least one example of what this...

This question might be elementary: If the norm of two complex numbers is equal, can we deduce that the two complex numbers are equal. I know in ℝ we can just look at this as an absolute value, but what about ℂ? So mainly: let |z| = |w|*|r| can we say → z = w*r ? Thanks

Hi, I want to show: \|f-jg\|^2 = \|f\|^2 - 2 \Im\{<f,g>\} + \|g\|^2 However, as far as I understand, for complex functions <f,g> = \int f g^* dt, right? Therefore: \|f-jg\|^2 = <f-jg, f-jg> = \int (f-jg)(f-jg)^* dt = \int (f-jg)(f+jg) dt = \int f^2 + jfg - jfg + g^2 dt = \|f\|^2 + \|g\|^2...

This is not really a homework question, but I've come across this while preparing for a test Homework Statement Let f:U \subseteq R^n -> R^m be a function which is differentiable at a \in U, and u \in R^n It is then stated that it is clear that: lim_{t \to 0}...

Homework Statement The Attempt at a Solution set x(t)=1+∫2cos(s(f^2(s)))ds(from 0 to t) then check x(0)=1+∫2cos(s(f^2(s)))ds(from 0 to 0)=1 then the initial condition hold, by FTC, we have dx(t)/dt=2cos(tx^(t)), then solutions can be found as fixed points of the map but for...

Homework Statement show that ||f||1 = ∫|f| (integral from 0 to 1) does define a norm on the subspace C[0,1] of continuous functions and also the same for ||f||= ∫t|f(t)|dt is a norm on C[0,1] Homework Equations (there are 3 conditions , i just don't know how to prove that...

Homework Statement find a norm on R2 for which||(0,1)||=1=||(1,0)|| but ||(1,1)||=0.000001 Homework Equations hints: ||(a,b)|| = A |a+b|+B|a-b The Attempt at a Solution by the hints i have A+B=1 and 2A=0.000001 then solved the equations system i get A=0.0000005...

http://imageshack.us/a/img141/4963/92113198.jpg hey, I'm having some trouble with this question, For part a) I know that in order for c_0 to be closed every sequence in c_0 must converge to a limit in c_0 but I am having trouble actually showing that formally with the use of the norm...

Prove that for $A \in \mathbb{R}^{n\times n} $ ||A||_{\infty} = \text{max}_{i=1,...,n} \sum_{j=1}^n |a_{ij} | I know that $||A||_{\infty} = \text{max} \dfrac{||Ax||_{\infty} }{||x||_{\infty}} $ such that $x \in \mathbb{R}^n$ any hints

Hello, I'm reading Gaussian measures on Hilbert spaces by S. Maniglia (available via google) and I have the following issue, regarding the proof. He states in Lemma 1.1.4: Let μ be a finite Borel measure on H. Then the following assertions are equivalent: (1) \int_H |x|^2 \mu(dx) < \infty (2)...

Hey Guys, I have tough dynamics question. I wasn't sure if this is the best place to post it, so if it isn't please let me know. I thought it might be because I couldn't find anything about it anywhere. So here goes. If one had 10 inch long hose, 1" in diameter, filled with some incompressible...

Homework Statement Prove that the Frobenius norm is indeed a matrix norm. Homework Equations The definition of the the Frobenius norm is as follows: ||A||_F = sqrt{Ʃ(i=1..m)Ʃ(j=1..n)|A_ij|^2} The Attempt at a Solution I know that in order to prove that the Frobenius norm is indeed...

Hi everyone, I need help with a derivation I'm working on, it is the differentiation of the norm of the gradient of function F(x,y,z): \frac{∂}{∂F}(|∇F|^{α}) The part of \frac{∂}{∂F}(\frac{∂F}{∂x}) is bit confusing. (The answer with α=1 is div(\frac{∇F}{|∇F|}), where div stands for...

Hi, All: I am working on a proof of the fact that any two norms on a f.dim. normed space V are equivalent. The idea seems clear, except for a statement that (paraphrase) any norm in V is a continuous function of any other norm. For the sake of context, the whole proof goes like this...

Can anyone explain Why l1 Norm is non-differentiable in terms of matrix calculus ?

Hello everyone! I came across, in a reading, an unfamiliar norm notation: $||A-B||_{2,a}$ where $a$ is the standard deviation of a Gaussian kernel. Now I know that the index 2 represents the $\ell _2$ norm, but what about the $a$? Moreover, is the matrix norm defnied in a similar way to the...

Homework Statement Show that the sup norm on R2 is not derived from an inner product on R2. Hint: suppose <x,y> is an inner product on R2 (not the dot product) and has the property that |x|=<x,y>0.5. Compute <x±y, x±y> and apply to the case x=e1, y=e2.Homework Equations |x|=<x,y>0.5 I've...

Homework Statement L : R^n → R is defined L(x1 , . . . , xn ) = sum (xj) from j=1 to n. The problem statement asks me to find an estimation for the operation norm of L, where on R the norm ll . llp, 1 ≤ p ≤ ∞, is used and on R the absolute value.The Attempt at a Solution from, ll Lv...

‖A(x+αz)-b‖_2^2 where A is an mxn matrix, x and z are vectors in R^n, b is a vector in R^m, and α is a scalar.

What conditions most be true for these two norms to be equal? Or are they always equal?

Homework Statement A(\vec{x}) = (F + T * x )2 F is a constant, x is a 2×1 vector T is a (constant) 1×2 matrixB(\vec{x}) = || K.Z.x ||2 k:3\times3 matrix and Z:3\times2, x the same as aboveB(x) is also R2→RC(x) = A(x) + B(x) Homework Equations 1- I am confused...