Norm Definition and 263 Threads

Homework Statement check whether ||.|| : \Re^{2} -> \Re_{+} given by (x,y) = |x| + |y|^{2} is a norm on R2 Homework Equations For all a in F and all u and v in V, 1. p(a v) = |a| p(v), (positive homogeneity or positive scalability) 2. p(u + v) ≤ p(u) + p(v) (triangle...

so i came up with a proof that..well.. Let L/K be a field extension and we have defined the norm and trace of an element in L, call it a, to be the determinant (resp. trace) of the linear transformation L -> L given by x->ax. Now it's well known that the determinant and trace are the...

I'm confused with the definition of a norm ordering of operators. The basic definition of norm ordering as understood by me was "Place the annihilation operators to right and creation operators to the left". However I also read another definition "The motivation of norm ordering is to ensure...

Homework Statement Let ||\cdot || denote any norm on \mathbb{C}^m. The corresponding dual norm ||\cdot ||' is defined by the formula ||x||^=sup_{||y||=1}|y^*x|. Prove that ||\cdot ||' is a norm. Homework Equations I think the Hölder inequality is relevant: |x^*y|\leq ||x||_p ||y||_q...

I was wondering about useful norms on tensor products of finite dimensional vector spaces. Let V,W be two such vector spaces with bases \{v_1,\ldots,v_{d_1}\} and \{w_1,\ldots,w_{d_2}\}. We further assume that each is equipped with a norm, ||\cdot||_V and ||\cdot||_W. Then the tensor product...

Could anyone tell me where to find a proof of the fact that the supremumnorm is a norm? The supremum norm is also known as the uniform, Chebychev or the infinity norm.

Homework Statement Consider the half space defined by H = {x ∈ IRn | aT x +alpha ≥ 0} where a ∈ IRn and alpha ∈ IR are given. Formulate and solve the optimization problem for finding the point x in H that has the smallest Euclidean norm. Homework Equations The Attempt at a...

Homework Statement a) Let V be a normed vector space. Then show that (by the triangle inequality) the function f(x)=||x|| is a Lipschitz function from V into [0,∞). In particular, f is uniformly continuous on V. b) Show that a closed subset F of contains an element of minimal norm, that...

Hello. My question is: does the norm on a space depend on the choice of basis for that space? Here's my line of reasoning: If the set of vectors V = \left\{ v_1,v_2\right\} is a basis for the 2-dimensional vector space X and x \in X, then let \left(x\right)_V = \left( c_1,c_2\right)...

Homework Statement Find two functions f, g \in C[0,1] (i.e. continuous functions on [0,1]) which do not satisfy 2 ||f||^2_{sup} + 2 ||g||^2_{sup} = ||f+g||^2_{sup} + ||f-g||^2_{sup} (where || \cdot ||_{sup} is the supremum or infinity norm) Homework Equations Parallelogram identity...

Homework Statement x[ Prove the continuity of the norm; ie show that in any n.l.s. N if xn \rightarrow x then \left|\left|x_n\left|\left| \rightarrow \left|\left|x\left|\left| The Attempt at a Solution i don't know where to start this from the definition of convergence xn \rightarrow x...

Homework Statement ||T|| = {max|T(x)| : |x|<=1} show this is equivalent to ||T|| = {max|T(x)| : |x| = 1} The Attempt at a Solution {max |T(x)| : x<=1} = {max ||x|| ||T(x/||x||)|| : |x|<=1} <= {max ||T(x)|| : |x| = 1} does that look right? I need to show equality...

Homework Statement I was just wondering how I would go about proving that the euclidean metric is always smaller than or equal to the taxicab metric for a given vector x in R^n. The result seems obvious but I am not sure how I would show this. Homework Equations The Attempt at a Solution

Homework Statement E1 = {pn(t) = nt(1-t)n:n in N}; E2 = {pn(t) = t + (1/2)t2 +...+(1/n)tn: n in N}; where N is set of natural numbers is the polynomial bounded w.r.t the supremum norm on P[0,1]? Homework Equations supremum norm = ||*|| = sup{|pn(t)|: t in [0,1]} The Attempt...

I just want to make sure I'm straight on the definition. Am I correct in assuming that, if I want to show that a sequence \langle f_n \rangle of functions converges to 0 in the L^1 norm, I have to show that, for every \epsilon > 0, there exists N \in \mathbb N such that \int |f_n| < \epsilon...

\int|x|2 with respect to the vector x in the unit ball in Rn-2 I'm dealing with volumes of unit balls in Rn and after applying a change of variable to the last 2 components and Fubini's Theorem, I get that integral and can't find a way to integrate it. Any help on this?

Hello! It is said that not every metric comes from a norm. Consider for example a metric defined on all sequences of real numbers with the metric: d(x,y):=\displaystyle\sum_{i=1}^{\infty}\frac{1}{2^i}\frac{|x_i-y_i|}{1+|x_i-y_i|} I can't grasp how can that be. There is a proof...

Hi, I've been trying to show that the set of matrices that preserve L1 norm (sum of absolute values of each coordinate) are the complex permutation matrices. Complex permutation matrix is defined as permutation of the columns of complex diagonal matrix with magnitude of each diagonal element...

Homework Statement \|x\|_2\le\|x\|_1\le\sqrt{n}\|x\|_2 where |x|1 is the l1 norm and |x|2 is the l2 normHomework Equations See aboveThe Attempt at a Solution I have \|\mathbf{x}\|_1 := \sum_{i=1}^{n} |x_i| and \|x\|_2 = \left(\sum_{i\in\mathbb N}|x_i|^2\right)^{\frac12} I have tried to...

norm of a function ||f|| & the "root mean square" of a function. How do I explain the connection between the norm of a function ||f|| & the "root mean square" of a function. You may like to consider as an example C[0,\pi], the inner product space of continuous functions on the interval [0,\pi]...

Hi all, Sorry if this is in the wrong section...first time and i couldn't see a convex analysis section. I'm trying to find a good algorithm/theorem that will maximise the l1 norm (sum of absolute values) of a linear function. Namely, given a function z = c + Ax where z is (nx1), A is...

Hello, I have a question regarding fft's. My experience with working with Fourier transforms is pretty much limited to transforming contrived functions pen and paper style. But now I need something and I think the fft is the appropriate tool, but I'm having a hard time understanding some...

Hello, I have a (infinite dimensional) vector space and defined an inner product on it. The vectors element are infinite sequence of real numbers (x_1, x_2,\ldots). The inner product has the common form: x_iy_i The problem now is that the vectors have an infinite number of elements, so the...

Let V be a vector space over the complex field. If V has an inner product <\cdot,\cdot>, and ||\cdot|| is the induced norm, then it's easy to show that the norm must satisfy the parallelogram law, to wit: ||x+y||^2 + ||x-y||^2 = 2||x||^2 + 2||y||^2 Much more interestingly, given an arbitrary...

Homework Statement Show that \frac{\Vert X(u+v) \Vert}{\Vert u+v \Vert} \leq \max \{ \frac{\Vert Xu \Vert}{\Vert u \Vert}, \frac{\Vert Xv \Vert}{\Vert v \Vert} \} Homework Equations The Attempt at a Solution Tried to rewrite the max statement as an inequality...

I thought that Jerry Springer shows was the tip of it all, but I guess this is the norm over there: http://news.yahoo.com/s/ap/20090501/ap_on_re_mi_ea/ml_saudi_child_marriage

Ok, so I have no idea how to take the norm of a vector composed of two vectors. I have \vec{q}=\vec{pi} - \vec{pf}we are given: |\vec{pi}|=|\vec{pf}|=|\vec{p}| so i know that |\vec{q}| \neq 0, that would be too easy, and it doesn't make sense. now, is the following right? it just doesn't seem...

Hi: I am trying to show that the ess sup norm is the limit of the L^p norms as p-->oo . i.e., ess sup =lim_p->oo ( {Int f^p)^1/p Please tell me if this is correct: 1) Def. ess sup f(t)=inf{M:m(t:f(t)>M)=0 } Then, f(t)>M only in the set S , with m(S)=0 , and f(t So Lim_p->oo...

Homework Statement X is the space of ordered n-tuples of real numbers and ||x||=max|\xij| where x=(\xi1,...,\xin). What is the corresponding norm on the dual space X'? Homework Equations The Attempt at a Solution I think the answer is that ||x*||=|x_1|+...+|x_n| , but I'm not sure...

Homework Statement From Calculus on Manifolds by Spivak: 1-7 A Linear Transformation T:Rn -> Rn is Norm Preserving if |T(x)|=|x| and Inner Product Preserving if <Tx,Ty>=<x,y>. Prove that T is Norm Preserving iff T is Inner Product Preserving. Homework Equations T is a Linear...

I'm trying to show that if f(x) is analytic, then for large enough n, || f^{(n)} (x) || \leq c n! || f(x) ||, where || f ||^2=\int_a^b{|f|^2}dx and f^{(n)} denotes the nth derivative. I tried to use the Taylor series, and then manipulated some inequalities, but I wasn't getting...

Euclidean norm is defined usually as|v|2= g(v,v), where g is a nondegenerate, positive definite, symmetric bilinear form. But how can make it backwards? What properties must norm have that g(v,w) = (|v+w|2 - |v|2 - |w|2)/2 be a positive definite, symmetric bilinear form?

I just want to verify if the following is correct \left\right\|x\|2.\left\right\|A\|2= \left\right\|Ax\|2 Thanks

Homework Statement http://img523.imageshack.us/img523/4456/56166304yr3.png Homework Equations http://img356.imageshack.us/img356/2793/40249940is8.png The Attempt at a Solution I defined K:[a,b] --> [a,b] with k(s,t) = \frac{(t-s)^{n-1}}{(n-1)!} I found for the norm: \int_a^b \int_a^b...

Homework Statement Is it possible to find the norm of a matrix? Not a column or row matrix which is a vector, but like on a 2x2 or 3x3 matrix? Homework Equations The Attempt at a Solution

Homework Statement I am trying to show that (1) ||A||_1 = \max_j \sum_i |a_{ij}| (2) ||A||_2 = \sqrt{max\_eigval\_ of\_{ } A^* A} where A* is the conjugate transpose (3) ||A||_\infty = \max_i \sum_i |a_{ij}| Homework Equations In general, ||A||_p = max_{x\neq 0}...

Homework Statement I'm stuck on this review problem for our final: The projection of X onto (a,b) = (a,b) X is orthogonal to (-a,b) Describe the norm of X in terms of a and b. The Attempt at a Solution I drew everything out on a Cartesian system, with the vector X being perpendicular to...

Homework Statement In R4, let U = span((1, 1, 0, 0), (1, 1, 1, 2)). Find u in U such that ||u - (1, 2, 3, 4)|| is as small as possible. Homework Equations The Attempt at a Solution I came up with a vector u = (-.5, -.5, 0, 0) + (2, 2, 2, 4) = (1.5, 1.5, 2, 4). Then u - (1, 2, 3, 4) = (0.5...

Suppose f(x)= -2x+1 is a vector in the vector space C[0,1]. Calculating the norm (f,f) results in 1/3. I'm a little confused. So on [0,1] the function is a straight line from (0,1) to (0,-1). So I thought I could simply takes this line segment and turn it into a directed line segment...

Homework Statement Let A = [a_{ij}] be a mxn matrix. Show that max_{ij}|a_{ij}| ≤ ‖A‖ ≤ √(∑_{ij}|a_{ij})|Homework Equations The Attempt at a Solution By the definition ‖A‖=max_{||x||≤1}‖A(x)‖ for all x ∈ Rⁿ.So, ‖A‖≥‖A∘(x₁,..,x_{n})^{T}‖ for x = (0,...,1,...0) with 1 is in the i^{ij} position...

Homework Statement (I'm posting this because my proofs seem to be lousy. I want to see if I'm missing anything.) Show that if f_n \in L^2(a,b) and f_n \rightarrow f in norm, then <f_n,g> \rightarrow <f,g> for all g \in L^2(a,b) Homework Equations L^2(a,b) is the space of...

norm is defined to be the length of the vector and we put we denote it by ||a||. However, modulus |a| also means the length of a from the origin? So, what is the difference between the symbol || || and | |?

I have the next problem. I have to proof that \left\vert x_{i}\right\vert\leq\left\vert\left\vert x\right\vert\right\vert \forall x\in\mathbb{R}^{n} with the usual scalar product and norm. It's obvious that x_{i}^{2}=\left\vert x_{i}\right\vert^{2}\leq \max\left\{\left\vert...

Homework Statement I need to prove that any two quadratic integers that are associates must also have the same norm. Homework Equations If α = a + b√d, the norm of α is N(α) = a^2 - b^2*d. If two quadratic integers α and β are associates, α divides β, β divides α, and α/β and β/α both equal...

I'm sure whoever is familiar with this subject has already seen this several times. I've seen it several times myself, and I even remember proving it in detail a couple of years ago, but now I'm stuck. I'm quoting what my professor did in class. Given some separable extension L/K, say for...

Q1: how do we find a vector x so that ||A|| = ||Ax||/||x||(using the infinite norm) totally no clue on this question.. Q2: Suppose that A is an n×n invertible matrix, and B is an approximation of A's inverse A^-1 such that AB = I + E for some matrix E. Show that the relative error in...

Suppose \mathbb{T}=[-\pi,\pi] and we have a function in L^p(\mathbb{T}) with some measure. If we know the Fourier coefficients of f, what is the L^p norm of f? Is it (\sum f_i^p)^{1/p}? where fi are the coefs.

Homework Statement Let F(AB) be the Frobenius-Norm in respect of the matrix A*B. And let ||A||2 be the operator norm. I have to show that F(AB)<=F(B)*||A||2 2. The attempt at a solution I wrote F(AB) in terms of sums and then tried to go on. But I don't know how I could include the...

This is a routine minimization problem, find the closest point or points to b = (-1,2)^T that lie on (a) the x-axis and (b) the line y=x. First I am supposed to solve it with the Euclidian norm, which is no problem, but then we are supposed to solve with the L_\infty norm. I am a little...

Alright, so if I want to find the distance of the ∞-norm between two vectors in lR^3, then would I take the max of the vectors first and then subtract, or should I subtract the vectors and then take the max? I think that the vectors are subtracted, and then the norm is taken, but I just want to...