What is Norm: Definition and 278 Discussions

Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the environment, such as uranium, thorium and potassium and any of their decay products, such as radium and radon. Produced water discharges and spills are a good example of entering NORMs into the surrounding environment. Natural radioactive elements are present in very low concentrations in Earth's crust, and are brought to the surface through human activities such as oil and gas exploration or mining, and through natural processes like leakage of radon gas to the atmosphere or through dissolution in ground water. Another example of TENORM is coal ash produced from coal burning in power plants. If radioactivity is much higher than background level, handling TENORM may cause problems in many industries and transportation.

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  1. V

    Infinity and one norm question

    Hi, I was wondering why the one and infinity norm of a complex vector x are not equal to the the one and infinity norm of x* (the conjugate transpose of x)? This seems to be true for the 2-norm, but I am not sure why for these other norms.
  2. A

    Uniform continuity and the sup norm

    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...
  3. C

    Understanding the Norm Inequality ||Av|| ≤ ||A||||v||

    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...
  4. S

    Finding Orthogonal Vectors in R4 with Norm 1

    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...
  5. Sudharaka

    MHB Norm of a Bounded Linear Functional

    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...
  6. M

    Calculating the norm of linear, continuous operator

    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...
  7. S

    Real Analysis: L∞(E) Norm as Limit of a Sequence

    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...
  8. twoski

    Infinity/One Norm Multiplication

    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}...
  9. Y

    Norm of a Matrix Homework: Show ||A|| $\leq$ $\lambda \sqrt{mn}$

    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...
  10. I

    Proving the Frobenius Norm Identity for Matrices

    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?
  11. S

    MHB Why Does Vector Norm Use "Double" Absolute Value?

    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...
  12. S

    (STATISTICS) 3 randomly selected observations from standard norm dist

    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...
  13. N

    Inequality: Integration of Norm

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

    Find Length of P in Vector Norm Given Plane

    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...
  15. H

    Exploring Invariant Matrix Norms: Understanding the Frobenius Norm and Beyond

    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
  16. M

    How to Estimate the Operator Norm ||A||_2 for a Difference Operator?

    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...
  17. H

    MHB Calculating an integral norm in L2

    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 )...
  18. P

    Why is L1 norm harder to optimize than L2 norm?

    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...
  19. A

    BMR and adaptive thermogenesis: what's the 'norm'?

    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...
  20. K

    Does least squares solution to Ax=b depend on choice of norm

    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...
  21. K

    Calculating Vector Norms: Solving for Magnitude of Vectors

    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 \|...
  22. B

    Proof that a given subspace of C[−1,1] with L2 norm is closed

    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...
  23. T

    MATLAB help, code for Frobenius norm

    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...
  24. Fernando Revilla

    MHB Solve Equation 5x - ||v|| v = ||w||(w-5x)

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

    Comparing Norms & Metrics: Axioms & Differences

    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)...
  26. Y

    Norm and Rows of Complex Matrix

    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...
  27. A

    Euclidean Norm of a Vector: Exploring Its Importance

    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...
  28. I

    MHB Proof about the continuity of a function of norm

    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 $x = 0$.
  29. Fernando Revilla

    MHB Chen's question at Yahoo Answers (continuity of the norm).

    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.
  30. E

    L2 norm for complex valued vector

    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)..?
  31. E

    Example please for matrix norm

    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...
  32. B

    Can We Assume Equality of Complex Numbers Based on Their Norm?

    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
  33. D

    What Causes the Error in Calculating the L2 Norm of Complex Functions?

    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...
  34. B

    How to go from limit of vector norm to 'normal' limit

    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}...
  35. C

    How Can Fixed Points Determine Solutions in Differential Equations?

    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...
  36. C

    Proving Integral Norms on C[0,1] for Continuous Functions

    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...
  37. C

    Find Norm on R2 with ||(0,1)||=1=||(1,0)|| & ||(1,1)||=0.000001

    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...
  38. G

    Show subspace of normed vector is closed under sup norm.

    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...
  39. A

    MHB What is the Maximum Norm Proof for Matrix A?

    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
  40. C

    How Does Trace Class Operator Q Relate to Norms in Hilbert Spaces?

    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)...
  41. S

    Beyond the norm Dynamics question

    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...
  42. T

    Proving the Frobenius Norm as a Matrix Norm

    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...
  43. W

    Differentiation of the l1 norm of gradient

    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...
  44. B

    Continuity of one Norm w.resp. to Another. Meaning?

    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...
  45. V

    Why l1 Norm is non-differentiable?

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

    MHB What does the notation $||A-B||_{2,a}$ represent in terms of matrix norms?

    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...
  47. C

    Sup norm and inner product on R2

    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...
  48. N

    Estimation of the Operator norm

    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 lly ≤...
  49. G

    How do you evaluate a norm like this?

    ‖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.
  50. G

    When is the Frobenius norm of a matrix equal to the 2-norm of a matrix?

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