Norm Definition and 263 Threads

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) -...

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

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...

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...

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...

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...

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...

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...

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)...

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

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...

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...

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...

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.

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

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...

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...

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

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...

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}...

"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...

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}||?

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...

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...

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. :)

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...

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...

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...

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...

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...

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...

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...

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...

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?

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...

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

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}...

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!

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!

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.

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...

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...

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

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?

Hello PhysicsForums! I was reading up on UFD's and I came up with a few quick questions. 1. Why don't the integers of Q[\sqrt{-5}] form a UFD? I was trying to tie in the quadratic integers that divide 6 to help me understand this, but I am stuck. 2. Why is Z a UFD? 3. Assuming Q[\sqrt{d}] is...

Hi all! I hope this is the right section to post such a question... I'm studying the theory of resolvent from the QM books by A. Messiah and I read in a footnote (page 713) that the norm of the resolvent satisfies \|R_A(z)\| = \lVert \frac{1}{A-zI} \rVert \ge \text{dist}(z,\sigma(A))^{-1}...

Homework Statement http://dl.dropbox.com/u/4027565/2010-10-10_194728.png Homework Equations The Attempt at a Solution ||B|| = ||M_1 * A * M_2 || So from an equality following from the norm, we can get... ||B|| <= ||M_1||*||A||*||M_2||. Now, we know that B is a...

Hello everyone, I need some help with finding norms of the field extension. I feel pretty comfortable when representing norms as determinants of linear operators but I seem to be stuck with representing norms as product of isomorphims. I have read Lang's GTM Algebra, but I really would...

Homework Statement The Attempt at a Solution I let D be the center x = DX & a = DA (x-a) * (x+a)=|x|^2-|a|^2 Dunno what to do with the right side of the equation

Hello, I know that the squared norm of a multivector M in a Clifford Algebra \mathcal{C}\ell_{n,0} is given by: <M \widetilde{M}>_0 that is the 0-grade part of the product of M and its grade-reversal. Is there a more general definition of squared-norm (for multivectors) that works...