Norm Definition and 263 Threads

Let ##(M,g)## a manifold with a Levi-Civita connection ## \nabla ## and ##X## is a vector field. What is the formula of ## | \nabla X|^2 ## in coordinates-form? I know that ##|X|^2= g(X,X)## is equivalent to ## X^2= g_{ij} X^iX^j## and ##\nabla X## to ##\nabla_i X^j = \partial_i X^j +...

I need help understanding this notation, what does this mean? Squared of 2-norm? 1. Homework Statement Thanks

Can someone give me in very abstract terms what a sobolev norm is or means?

Homework Statement Find coefficients a,b>0 such that a||x||∞≤||x||≤b||x||∞.Homework EquationsThe Attempt at a Solution No idea how to get started. Help will be appreciated.

Hi. I read that the Lorentz invariance Minkowski norm of the four-momentum $$E^2-c^2\cdot \mathbf{p}^2=m^2\cdot c^4$$ has no analogue in Newtonian physics. But what about $$E-\frac{\mathbf{p}^2}{2m}=0\quad ?$$ It might look trivial by the definition of kinetic energy, but it's still a relation...

Sorry I'm a little rusty with my math and proof logic, and this feels like a dumb question, but oh well! The Euclidian norm of a vector in ℝ3 is \|{v}\| = \sqrt{x^2 + y^2 + z^2} where \|{v}\| \geq 0. I'm trying to show that there is always an infinite number of solutions for arbitrary...

Homework Statement Hi everybody! Here is another problem about contraction and Banach fixed-point theorem that I don't get: The function ƒ: C([0,½]) → C([0,½]) is defined by: [f(x)](t) := 1 + \int_{0}^{t} x(s) ds ∀ t∈[0,\frac{1}{2}]. Is ƒ a contraction with respect to the norm || ⋅ ||∞? If...

Homework Statement Hi everybody! I have a math problem to solve, I'd like to check if I understand well the Banach fixed-point theorem in the case of Euclidean norm and how to deal with maximum norm. Check if the following functions ƒ: ℝ2 → ℝ2 are strictly contractive in relation to the given...

Homework Statement Hi there. I have to prove this inequality: ##||x||_2 \leq ||x||_1 \leq \sqrt{n} ||x||_2## Where ##||x||_2## is the ##l_p## norm with p=2, so that: ##||x||_2=(|x_1|^2+|x_2|^2+...+|x_n|^2)^{\frac{1}{2}}## And similarly ##||x||_1=|x_1|+|x_2|+...+|x_n|## is the ##l_1##...

Homework Statement The weighted vector norm is defined as ##||x||_W = ||Wx||##. W is an invertible matrix. The induced weighted matrix norm is induced by the above vector norm and is written as: ##||A||_W = sup_{x\neq 0} \frac{||Ax||_W}{||x||_W}## A is a matrix. Need to show ##||A||_W =...

Homework Statement Prove $$||UA||_2 = ||AU||_2$$ where ##U## is a n-by-n unitary matrix and A is a n-by-m unitary matrix. Homework Equations For any matrix A, ##||A||_2 = \rho(A^*A)^.5##, ##\rho## is the spectral radius (maximum eigenvalue) where ##A^*## presents the complex conjugate of A. U...

I am trying to prove ##||A||_{\infty} = max_i \sum_{j} |a_{ij}|## which reads as the ##\infty## norm is the max row sum of matrix A. ##i## is the row index and ##j## is the column index. Here is what I thought of: ##||A||_{\infty} = sup_{x\neq 0} \frac{||Ax||_{\infty}}{||x||_{\infty}}## The...

The induced matrix norm for a square matrix ##A## is defined as: ##\lVert A \rVert= sup\frac{\lVert Ax \rVert}{\lVert x \rVert}## where ##\lVert x \rVert## is a vector norm. sup = supremum My question is: is the numerator ##\lVert Ax \rVert## a vector norm?

I have attached a derivation of the Minkowski norm of the four-momentum but just don't quite see how the writer arrived at ## -m^2 c^2 ## from what was given. How exactly does this quantity follow from ## -\frac {E^2}{c^2} + p^2##? I feel like it might be very obvious, so any explanation would...

Hello : CONTEXT : let be E the space of rapidly decreasing functions on ##\mathbb{R}^{n}## in ##\mathbb{R}##. I define $$(||.||_{i})_{i \in \mathbb{R}^{n+1}}$$ with forall ##i = (k, m_{1},\ldots, m_{n}) = (k, m)## we define for ##f## in ##E## ##||f||_{i} = \sup_{x \in \mathbb{R}^{n}} \Big|(1...

Assuming a four velocity ##u_{\nu}=(1,0,0,0)## we can use the Maxwell Energy-Momentum Tensor to build a 4-vector in the following way ##P^{\mu}=u_{\nu}T^{\mu\nu}=\left(\frac{E^{2}+B^{2}}{2},\mathbf{E}\times\mathbf{B}\right)## So, we have a vector whose time component is the energy density of...

I am to study how fast an iterative method for nonlinear system of equations converges to a certain root and found out that I can evaluate my rate of convergence by using the following formula: ##r^{(k)}=\frac{||x^{(k+1)}-x^{(k)}||_V}{||x^{(k)}-x^{(k-1)}||_V}##. My question is which vectorial...

If a sequence of operators \{T_n\} converges in the norm operator topology then: $$\forall \epsilon>0$$ $$\exists N_1 : \forall n>N_1$$ $$\implies \parallel T - T_n \parallel \le \epsilon$$ If the sequence converges in the strong operator topology then: $$\forall \psi \in H$$...

Homework Statement Let |v> ∈ ℂ^2 and |w> = A|v> where A is an nxn unitary matrix. Show that <v|v> = <w|w>. Homework Equations * = complex conjugate † = hermitian conjugate The Attempt at a Solution Start: <v|v> = <w|w> Use definition of w <v|v>=<A|v>A|v>> Here's the interesting part Using...

I have a linear integral operator $K\psi=\int_{a}^{b} \,k(x,s) \psi(s) ds$ $L\psi=\int_{a}^{b} \,l(x,s) \psi(s) ds$ both are continuous I know how to obtain the eigenvalues of each alone. But how can I calculate the eigenvalues of the operator $I-{L}^{-1} K$ or at least the norm...

Homework Statement Find an example of a sequence ##\{ f_n \}## in ##L^2(0,\infty)## such that ##f_n\to 0 ## uniformly but ##f_n \nrightarrow 0## in norm. Homework Equations As I understand it we have norm convergence if ##||f_n-f|| \to 0## as ##n\to \infty## and uniform convergence if there...

Suppose we pick a matrix M\in M_n(ℝ) s.t. all its eigenvalues are strictly bigger than 1. In the question here the user said it induces some norm (|||⋅|||) which "expands" vector in sense that exists constant c∈ℝ s.t. ∀x∈ℝ^n |||Ax||| ≥ |||x||| . I still cannot understand why it's correct. How...

Hi, I found a statement without a proof. It seems simple enough, but I am having trouble proving it because I am not positive about induced matrix norms. The statement is that $$||A^k|| \leq||A||^{k}$$ for some matrix A and positive integer k. I have found that the norm of a matrix is the...

Why is it that only one significant figure for any uncertainty is taught? It seems like a nice general rule, but isn't it an unnecessary constraint and ultimately a poor rule in many cases? For example ## 5 \pm 1 ## could mean an error of either 0.5 or 1.5, which is a very large range. Wouldn't...

I am reading an article[1] that states: Let k be a fixed local field. Then there is an integer q=pr, where p is a fixed prime element of k and r is a positive integer, and a norm |.| on k such that for all x∈k we have |x|≥0 and for each x∈k\{0} we get |x|=qm for some integer m. This norm is...

I have a linear integral operator (related to integral equations) $(Ky)(x)=\int_{a}^{b} \,k(x,s)y(s)ds$ If $|b|. ||K||<1$ (b is a scalar) can I say $||(I-bK)-1||< 1 / (1-|b|.||K||)$ I think it's correct is it? thanks

Hello! (Wave) Let $V=C^1([a,b])$. Show that if $J$ is a continuous functional in respect to the norm $||y||_1:=||y||_{\infty}+||y'||_{\infty}, y \in V$ then it is also continuous in respect to the norm $||y||:=||y||_{\infty}$. Also, show that the inverse of the above claim does not hold. Let...

Hi, For the past couple of days I've been attempting to derive the equality (for any normalised ##\varphi##) ||(a+za^\dagger+ \lambda)\varphi ||^2 = ||(a^\dagger+\bar{z}a+ \bar{\lambda} )\varphi ||^2 + (|z|^2-1)||\varphi ||^2 First of the summation seemed like a typo. So I first tried to prove...

Can I always say without reservation that for any two integral operators $K$ and $L$ defined as follows say $(Ky)(x)=\int_{a}^{b} \,k(x,s)y(s)ds$ that $||L||+||K-L||\ge||K||$ thanks Sarrah

Hi I have 2 linear integral operators $(Ku)(x)=\int_{a}^{b} \,k(x,t)u(t)dt$ $(Mu)(x)=\int_{a}^{b} \,m(x,t)u(t)dt$ I am defining $||K||=max([x\in[a,b]\int_{a}^{b} \,|k(x,t| dt$ same for $L$ when does $||K+M||=||K||+||M||$ thanks sarrah

i have a problem concerning the norm of linear integral operator. ii found the answer in a book called unbounded linear operators theory and applications by Dover books author seymour Goldberg. the proof runs as follows ||T|| is less than max over x in [a,b] of integral (|k(x,y)|dy) Then he...

In this book I'm working from (P-adic numbers: An introduction by Fernando Gouvea), he gives an explanation of the p-adic expansion of a rational number which I'm pretty sure I understand this, but a bit later he talks about the p-adic absolute value, which from what I understand is the same as...

So I'm studying norm of matrices in my calc class, and most resources I've looked at seem like it's just the square root of all the entry values over the sum of all the entries, but when given the matrix [1 3] [0 1] Sqrt((11+sqrt(117))/2) The 11 is the sqrt of 12 + 32 + 12 the 2 is the root...

Homework Statement When is |x+y|=|x|+|y| for arbitrary non-zero vectors x,y∈Rn ie, when does equality hold for the well known inequality |x+y|≤|x|+|y| Homework Equations |x|2=<x,x>=Σixi2 |x+y|≤|x|+|y| The Attempt at a Solution squaring both sides of the inequality we have...

If we have a vector $$\partial_v$$ and we want o find its norm, we easily say (According to the given metric) that the norm of that vector is:$$ g^{vv}\partial_v\partial_v$$. My question what if we have a vector that is combination of 2 vectors like: $$\phi =\partial_v + a\partial_x$$ where $a$...

Let $$(X, d)$$ be a metric space, $$AE_0(X) = \{ u : X \rightarrow \mathbb{R} \ : \ u^{-1} (\mathbb{R} \setminus \{0 \} ) \ \ \text{is finite}, \ \sum_{x \in X} u(x)=0 \}$$, for $$x,y \in X, \ x \neq y, \ m_{xy} \in AE_0(X), \ \ m_{xy} (x)=1, \ m_{xy}(y)=-1, \ m_{xy}(z)=0$$ for $$z \neq x...

Hey! :o If $f_n, f \in L^p, 1\leq p < +\infty$ and $f_n \rightarrow f$ almost everywhere, and $||f_n||_p \rightarrow ||f||_p$, then $f_n\rightarrow f$ as for the norm. Could you give me some hints how to show it?? (Wondering) What does convergence as for the norm mean?? (Wondering)

Hey! :o In $\mathbb{R}^d$ with the Lebesgue measure if $f \in L^p, 1 \leq p < +\infty$, and if for each $y$ we set $f_y(x)=f(x+y)$ then: $f_y \in L^p$ and $||f||_p=||f_y||_p$ $\lim_{y \rightarrow 0} ||f-f_y||_p=0$ Could you give me some hints how to show that?? (Wondering)

Hi folks, 1. Homework Statement I don't fully understand the question statement, how is it supposed to be read? Question: Give a formula for the minimizer x* (to be read as x-star) of the function ƒ:ℝn → ℝ, x → ƒ(x) = ||Ax-b||22, where A∈ℝm×n and b∈ℝm are given. You can assume that A has rank...

If $\|(x,y)\|=\sqrt{x^2+4y^2}.$ Show that $\|.\|$ is an norm. Hello MHB! :) The cases $\|(x,y)\|\ge 0$ and $\|\lambda (x,y)\|=|\lambda|\|(x,y)\|$ are obvious, but the case $\|(x,y)+(x',y')\|\le \|(x,y)\|+\|(x',y')\|$ I don't know how it do? Help me!

Homework Statement Homework Equations How can I start the proof? Shall I use the Poincare inequality? The Attempt at a Solution Well, I know that this norm is defined by , but still I don't know how to start constructing the proof?

Hi guys, I've attached a problem that I've been struggling with for a while now. I was wondering if anyone had some advice on how to approach it (in particular part a) or some resources they could recommend to me?Thanks in advance, Sam

Homework Statement In the previous few modules you studied the problem of minimizing ##\| Ax -b \|_{2}## by choice of ##x##. So far you've done this in Matlab using either the backslash operator or the command pinv. Now that you've been exposed to linear programming, you have the tools to solve...

Hi I have an integral over [0,1] of product of two matrices say A(t). B(t) and I wish to bound its norm. Can you say that ||integral (AB)||<||B(t)||.||integral (A)|. is there some conditions on that to occur thanks sarrah

I'm working with some old software to optimize antenna networks, and I've come across some stuff that I don't understand. For the software to run, all impedence values entered must be normalized to 1 ohm. What does it mean to normalize an impedence and how do I do it? Also, the manual for the...

How can I calculate the following matrix norm in a Banach Space: $$ A=\begin{pmatrix} 5 & -2 \\ 1 & -1 \\ \end{pmatrix} ?$$ I have tried $$\|A\|=\sup\limits_{\|x\|=1}\|Az\|$$ and then did $$Az=\begin{pmatrix} 5 & -2 \\ 1 & -1 \\...

Hi PF people! I am not sure my question can elegantly fit in the template, but I 'll try. Homework Statement I am self-studying the 8th chapter of "Mathematical Methods for Physics and Engineering", 3rd edition by Riley, Hobson, Bence. In the section about unitary matrices, it is stated that...

I've seen a few references to it, but can't find it defined anywhere. What is it? Something with integrating distances between measures and something. Pushforward measure or something else?

I am a bit confused, so this question may not make much sense. A unitary operator from one vector space to another is one whose inverse and Hermitian transpose are identical. It can be proved that unitary operators are norm preserving and inner product preserving. Which raises the 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.