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. S

    Weird Sum of Squares as a Vector Norm and Gauss-Newton optimization

    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...
  2. S

    Wierd sum of squares compilation to a single vector norm

    I've encountered a function like this: S(x) = [M(x) - F(x)] ^2 + || G(x) || ^ 2X being a 3*1 vector M and F: vector----->scalar G: vector------->vector and || G || meaning its norm To change S(x) into a single square, authors have described it like this: S(x) = || A + B || ^ 2 where A=(M(x) -...
  3. J

    Solving Vector Norm with F Matrix: Advice from Jo

    Hi guys Assume F to be a square matrix, say 3 by 3. Now I want to find a vector q (3 by 1) to meet the requirement that norm(F*q)=1. How can I find it? What is the solution in general? THanks in advance! Jo
  4. A

    Convergence in L^2 Norm: Understanding Subsequence Implications

    Suppose there exists a sequence f_n of square-integrable functions on \mathbb R such that f_n(x) \to f(x) in the L^2-norm with x \ f_n(x) \to g(x), also in the L^2-norm. We know from basic measure theory that there's a subsequence f_{n_k} with f_{n_k}(x) \to f(x) for a.e. x. But my professor...
  5. B

    Space l_infty(R) with sup norm

    Homework Statement Prove that the space l_\infty (R) of bounded sequences with the sup norm (ie x=(x_n) in l_\infty (R) )is not a inner product space. The Attempt at a Solution Using definition of parallelogram ||x+y||^2+||x-y||^2=2(||x||^2+||y||^2) (1) Consider x_n=1^-n and...
  6. P

    Understanding the weak norm and it's notation

    Hello. I'm trying to grasp the notation for the definition of something called the weak q-norm, defined as \|x\|_{q,w}^q = \sup\limits_{\epsilon > 0} \epsilon^q \left| \Big\{i \,|\, |x_i| > \epsilon \Big\} \right| I don't come from a pure math background so I've never seen this...
  7. O

    Norm induced by inner product?

    On a finite-dimensional vector space over R or C, is every norm induced by an inner product? I know that this can fail for infinite-dimensional vector spaces. It just struck me that we never made a distinction between normed vector spaces and inner product spaces in my linear algebra course...
  8. P

    Showing a Norm is not an Inner Product

    Show the taxicab norm is not an IP. taxicab norm is v=(x_{1}...x_{n}) then ||V||= |x_{1}|+...+|x_{n}|) I was thinking about using the parallelogram law but I would get this nasty...
  9. F

    Calculating the norm of an ideal in Z[√6]

    For part i) I deduced via Dedekind's criterion that <2> = <2,√6>2 & <3> = <3,√6>2 So ii) I am trying to do now, and my argument is thus: Let a be an ideal in Z[√6]. Suppose that N(a) = 24. By a proposition in my notes we have that a|<24> = <2,√6>6<3,√6>2 so a = <2,√6>r<3,√6>s for some r...
  10. F

    Calculating Norm of Prime Ideal p = (3, 1 - √-5)

    I need to calculate the norm of the ideal p = (3, 1 - √-5) All the information I have is that it's a prime ideal. I managed to calculate the normal of the ideal q = (3, 1 + √-5) (which was 3) by finding a the determinant of a base change matrix by considering an integral basis Here...
  11. J

    Is the Norm of Four-Acceleration Always Equal to Proper Frame Acceleration?

    I saw that the norm of four acceleration is equal to the magnitude of proper frame's acceleration. So, if the observer moves in x direction, following equation about norm of it's 4 acceleration is like that -(d^2 t / dτ^2) + (d^2 x / dτ^2) = d^2 x / dt^2 In comoving frame(proper frame)...
  12. B

    Completeness of R^2 with sup norm

    Homework Statement Given that R is complete, prove that R^2 with the sup norm is complete Homework Equations The Attempt at a Solution How may I tackle this? Thanks
  13. mnb96

    Problem with minimizing the matrix norm

    Hello, I have to to find the entries of a matrix X\in \mathbb{R}^{n\times n} that minimize the functional: Tr \{ (A-XB)(A-XB)^* \}, where Tr denotes the trace operator, and * is the conjugate transpose of a matrix. The matrices A and B are complex and not necessarily square. I tried to...
  14. J

    Calculating a metric from a norm and inner product.

    I typed the problem in latex and will add comments below each image. The supremum of |1 - x| seems dependent on the interval [a, b]. For instance, if [a, b] = [-500, 1], then 501 is the supremum of |1 - x|. But if [a, b] = [-1, 500], then 499 is the supremum of [1 - x]. So what should I...
  15. C

    Complex Convergence with Usual Norm

    Homework Statement Determine whether the following sequence {xn} converges in ℂunder the usual norm. x_{n}=n(e^{\frac{2i\pi}{n}}-1) Homework Equations e^{i\pi}=cos(x)+isin(x) ε, \delta Definition of convergence The Attempt at a Solution I would like some verification that this...
  16. B

    A working example wrt the supremum norm

    Folks, Could anyone give me a working example of a sequence of functions that converges to a function wrt to the supremum norm? Thank you.
  17. J

    How can we proof this matrix norm equality?

    ||A-1|| = max ||x|| / ||Ax|| x\inℝn, x≠0 . x is a vector.
  18. K

    Norm equivalence between Sobolev space and L_2

    Hello! I've found this paper, wherein page 33 states that the reverse Poincaré inequality gives \forall v \in H^1_0(\Omega) , \|v\|_{L^2(\Omega)} \leq C(\Omega) \|\nabla v\|_{L^2(\Omega)} This I can follow - it gives a norm equivalence between the norm of a vector and the gradient of its...
  19. W

    Whats the most common roller chain norm for bicycles?

    I am designing a bike in Autodesk Inventor for a university project, and I am stuck with the sprockets. Inventor can create them fairly easily when you know the norm of the sprocket and the number of teeth it has, but I don't know the standard of the sprocket I have to design; I merely know that...
  20. C

    Dual Norm Spaces (isomorphism/isometric)

    Homework Statement If X and Y are normed spaces, define \alpha : X^* x X^*\rightarrow (X x X)^* by \alpha(f,g)(x,y) = f(x)+g(y). Then \alpha is an isometric isomorphism if we use the norm ||(x,y)|| = max(||x||,||y||) on X x Y, the corresponding operator norm on (X x Y)^*, and the norm...
  21. P

    Norm of V in ℂ^n Using Inner Product

    Using the standard inner product in ℂ^n how would I calculate the norm of: V= ( 1 , i ) , where this is a 1 x 2 matrix
  22. A

    What is the Infinity Norm & Why Use It?

    Hi I was wondering about the meaning of the infinity norm || x ||_\inf= max\{|x_1|, |x_2|...|x_n| \} if a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, why do we assign the maximum (or sup) as the value of this norm ? It must be a...
  23. R

    Show the norm ||x|| is less or equal to A|x| for some constant A

    Homework Statement Show that \|x\| \leq A|x| \forall x \in \mathbb{R}, where A \geq 0. Homework Equations We know the norm is a function f: {\mathbb{R}}^{d} \to \mathbb{R}, such that: a) f(x) = 0 \iff x = 0, b) f(x+y) \leq f(x) + f(y), and c) f(cx) = |c|f(x) \forall c \in \mathbb{R}...
  24. D

    1-norm is larger than the Euclidean norm

    "1-norm" is larger than the Euclidean norm Define, for each \vec{x} = (x_1, \ldots, x_n) \in \mathbb{R}^n, the (usual) Euclidean norm \Vert{\vec{x}}\Vert = \sqrt{\sum_{j = 1}^n x_j^2} and the 1-norm \Vert{\vec{x}}\Vert_1 = {\sum_{j = 1}^n |x_j|}. How can we show that, for all \vec{x} \in...
  25. A

    Why is the term 'norm' used instead of 'absolute value' in vector spaces?

    I saw some books and say that norm is the absolute value in vector. If it also means absolute value, why don't we use absolute value |\vec{v}| instead we use ||\vec{v}||?
  26. R

    How to Minimize the infinity norm of a matrix function

    Hi , I have been thinking of this question for a long time. Can someone give me an advice? There are three known matrices M, N, and K. M is a (4*4) matrix: M= [ 1 0 2 3; 2 1 3 5; 4 1 1 2; 0 3 4 3 ] N is a (4*3) matrix: N= [ 3 0 4; 1 5 2; 7 1 3; 2 2 1 ] K is a...
  27. R

    Intuitive Explanation of What a p norm is, for any arbitrary p>0

    Supposing V is a normed vector space, the p-norm of {\bf x} \in V is: \lVert {\bf x} \rVert_p := \left(\sum_{i=1}^n |x|^p \right)^{\frac{1}{p}} There are 3 special cases: p= 2: Euclidean distance - 'as the crow flies' p = 1: Taxicab distance - sum the absolute value of components...
  28. L

    Notation in linear algebra and rule for square of matrix norm

    Hi. I have a few simple questions. (<- sorry, please click this image.) 1. What does the notation in the red circle mean? 2. Is there a rule for expanding square of norm? (e.g. || A*B*C ||^2) I don't really understand how the first eq. changes to the second eq. Thanks. :)
  29. L

    Bounding the L-Infty Norm of a Diffble Fn

    Hello, I would appreciate any assistance with the following question: Suppose f \in C^2[-1,1] is twice continuously differentiable. Prove that |f'(0)|^2 \leq 4 ||f||_\infty (||f||_\infty + ||f''||_\infty), where ||f||_infty is the standard sup norm. At first I thought Taylor expansion...
  30. O

    What is the Relationship Between Vectors and Normed Linear Spaces?

    I'm trying to do a problem concerning converging sequences in normed linear spaces. Can anyone help me prove that if x=(x1,x2...,xn) is a vector in an n dimensional vector space then |xi| where i=1,2...,n; is always less than or equal to ||x|| (norm of x). Maybe start out by writing x as a sum...
  31. O

    Prove C[0,1] is closed in B[0,1] (sup norm)

    So basically, my metric space X is the set of all bounded functions from [0,1] to the reals and the metric is defined as follows: d(f,g)=sup|f(x)-g(x)| where x belongs to [0,1]. I want to prove that the set of all discontinuous bounded functions, D[0,1] in X is open. My attempt - Start with an...
  32. Z

    What is the meaning of the norm of a linear functional?

    Hi everyone, I have been studying "Optimization by Vector Space Methods", written by David Luenberger and I am stuck in an obvious point at first glance. My problem is in page 105, where the norm of a linear functional is expressed in alternative ways. The definition for the norm of a linear...
  33. S

    Show that ||x|| is a norm on R^n

    Homework Statement Show ||x|| = \sqrt{x \cdot x} is a norm on \mathbb{R}^n. Homework Equations Prop. 1. ||x|| = 0 IFF x=0. 2. \forall c \in \mathbb{R} ||cx|| = |c| \, ||x||. 3. ||x+y|| \leq ||x|| + ||y||. Cauchy-Schwarz Inequality. The Attempt at a Solution Just want to...
  34. F

    Minimizing infinity norm squared

    I have to minimize an expression of the following type: min <a,x>-L||x-u||_inf^2 s.t.: ||x||_inf <= R, where a is a vector of coefficients, x is the vector of decision variables, <.,.> denotes the scalar product, R and L are scalars, u is some constant (known) vector, and 'inf' denotes...
  35. L

    What is the Norm of a Vector and How is it Used in Linear Algebra?

    Hello, I am studying for an exam in Linear Algebra. My teacher gave us an outline of things that we need to know and one of them is this: Find the norm of a vector v in n-dimensional space. Use it to find a unit vector in the same direction as v. I was just hoping someone might be able to...
  36. M

    Norm of an Operator: Show llTll = max ldl

    Homework Statement Let D be a nxn diagonal matrix and T:Rn -> Rn be the linear operator associated with D. ie., Tx = Dx for all x in Rn. Show that: llTll = max ldl where d1, ..., dn are the entries on the diagonal of DHomework Equations the smallest M for which llTxll <= M*llxll is the norm...
  37. A

    Norm of operator vs. norm of its inverse

    Are there any circumstances under which we can conclude that, for an invertible, bounded linear operator T, \| T^{-1} \| = \frac{1}{\| T \|} ? E.g., does this always hold if we know the inverse is bounded?
  38. S

    What is the \|f\|_{C^{1}} Norm?

    So, I'm working my way through a proof, which has been fine so far, except I've hit a bit of notation which has stumped me. Essentially, I have a diffeomorphism f: \mathbb{R}^{n} \to \mathbb{R}^{n} (in this case n = 2, but I assume that's fairly irrelevant), and I have the following norm: \| f...
  39. Demon117

    One-sided derivative of a norm function exists

    Homework Statement Suppose that || || is a norm on R^n. If p,v are in R^n, show that the one sided derivative lim( [||p+tv||-||p||]/t, t-->0+) exists. The Attempt at a Solution Letting q(t) = ||p+tv||-||p||/t, for s<=t in R, I have already shown that q is bounded below by -||v||...
  40. S

    Understanding the Surjectivity of the Norm Function in Finite Fields

    If we have N:F_q^n ...> F_q , be the norm function . can anyone explian how the map N is surjective .
  41. A

    Equivalent definitions for the norm of a linear functional

    Can someone please explain why the following three definitions for the norm of a bounded linear functional are equivalent? \| f \| = \sup_{0 < \|x\| < 1} \frac{|f(x)|}{\| x \|}, and \| f \| = \sup_{0 < \| x \| \leq 1} \frac{|f(x)|}{\| x \|}, and \| f \| = \sup_{\| x \| = 1}...
  42. T

    The notation of the norm of polynomials

    Homework Statement attached Homework Equations The Attempt at a Solution what is x_i? is it the coefficient of x or simply add up 1-5? i found the notation different from http://mathworld.wolfram.com/PolynomialNorm.html so i am confused. Thx!
  43. P

    Why is Supremum of a.u Less Than or Equal to r||a||_2?

    Consider a and u are vector of n entries, why the supremum of a dot u subject to the 2-norm of u is less than or equal to r equals r times 2-norm of a, i.e. sup{a.u | ||u||_2 <=r} = r ||a||_2? How can I work out that? Thank you!
  44. U

    Finding the Minimal L1 Norm Solution for Ax=b: A Linear Programming Approach

    Hi, Can anyone tell me how to find the minimal L1 norm solution to the problem Ax=b using a linear programming method possibly the simplex search?? Any links where I can find something ?? Khan.
  45. G

    Proving Another Vector Norm on C[0,1]

    Homework Statement does the function \| \|: C[0,1] \rightarrow R defined by \|f \|= |f(1)- f(0)| define a norm on C[0,1]. if it does prove all axioms if not show axiom which fails The Attempt at a Solution i don't really understand the question. i know the 4...
  46. M

    The enaquality of the norm of the matrix e^(tA)

    Homework Statement Find the largest possible \alpha and the smallest possible \mu, so that for every t>0 e\alphat\leq IIetAII \leqe\mut when A=( 0 1) ----------(-1 -1) Homework Equations The inequality above always holds true when \alpha=\alpha(A) and \mu=\mu(A) The Attempt at...
  47. B

    What Does the Norm of a Jacobian Matrix Represent?

    In "Differential Equations, Dynamical Systems and Introduction to Chaos", the norm of the Jacobian matrix is defined to be: |DF_x| = sup |DF_x (U)|, where U is in R^n and F: R^n -> R^n and the |U| = 1 is under the sup. ...|U| = 1 DF_x (U) is the directional derivative of F in the direction of...
  48. M

    Difference between a convex norm and strong convex norme ?

    hi :) if someone have any idea ? what is the difference between a convex norm and strong convex norme ?
  49. D

    When is the norm of a state equal to 1?

    My textbook says that all physical vectors in the quantum mechanical vector space are unit vectors. But elsewhere, there are quantities like <a|a> which are not assumed to be equal to 1. Why the discrepancy, and under what situations does a state have/not have norm 1?
  50. D

    If A(x,y) is a Positive Definate Bilinear Form then sqrt(A(x,x)) defines a norm

    Homework Statement The Problem is from Mendelson Topology. Let V be a vector field with the real numbers as scalars. He defines a bilinear form as a function A:V x V -> R s.t for all x,y,z an element of V and real numbers a,b,c A(ax +by, z) = aA(x,z) + bA(y,z) and A(x,by + cz) = bA(x,y) +...
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