Functional Definition and 416 Threads

Homework Statement Let e_{n}(t)= \frac{1}{ \sqrt{2\pi}}\cdot e^{int} for n\in\mathbb{Z} and -\pi\le t\le\pi. Show that for any f\in L^{2}[-\pi,\pi] we have that (f,e_{n})=\int_{-\pi}^{\pi}f(t)\cdot e^{-int}dt\to0 as |n|\to \infty. The Attempt at a Solution I want to use dominant convergence...

Homework Statement Given the structure of Alantolactone, find two functional groups. 2. The attempt at a solution This was a question that was on my exam recently. I answered Ester and Ether, however Ether was marked incorrect. Instead, only the answers Ester and Alkene were accepted. How is...

Hi some one please help me with the following problem Suppose that T_0 is the interior of a triangle in R^2 with vertices A,B,C. If T_1 is the interior of the trianlge whose vertices are midpoints of the sides of T_0, T_2 the intrior of the triangle whose vertices are midpoints of sides of...

Suppose X is a normed space and X*, the space of all continuous linear functionals on X, is separable. My professor claims in our lecture notes that we KNOW that X* contains functionals of arbitrarily large norm. Can someone explain how we know this, please?

Homework Statement problem didn't state, but I assume let V be a vector space: V = C^3 and scalar is C Homework Equations Define a non-zero linear functional T on C^3 such that T ((1, 1, 1)) = T ((1, 1, −1)) = 0 The Attempt at a Solution So let X1 = (1, 1, 1); X2 = (1, 1, -1); It...

hello everyone! I had a stuck in solving problem for a week now, so need help. Please help! the problem is as follows.In a closed interval I=[0,\pi], the 2-times continuously differentiable function \phi(x) and \psi(x) meet the following conditions (they're ranged in \mathbb{R}). \psi...

I have to choose a total of 12 modules for my 3rd year. I've everything decided except four of them. I want to eventually do research either General Relativity, quantum mechanics, string theory, something like that. I'm torn between Group Representations, with one of Practical numerical...

I tried a special edition that came with an electronics book and it had lots of problems. Then I downloaded a 9.1 student version that I found online but when I try to open a new sim profile it doesn't allow me to put any name I want or import. And the Capture Student version that I have doesn't...

http://latex.codecogs.com/gif.latex?\hspace{-20}$%20A%20function%20$f:\mathbb{N}%20\rightarrow%20\mathbb{N}$%20and%20satisfies%20$f(ab)%20=%20f(a)+f(b)$.\\%20Where%20$a$%20and%20$b$%20are%20Coprime%20Natural%20no.\\%20and%20$f(c+d)%20=%20f(c)+f(d)\forall$%20prime%20no.%20$c$%20and%20$d$.%20Then\\...

What is the relationship between topology, functional analysis, and group theory? All three seem to overlap, and I can't quite see how to distinguish them / what they're each for.

Hello everyone! I'm a bit confused about referring to a mapping as function or functional, for example: $f(x_1, x_2, x_3) = x_1+2x_2 ^2+3x_3 ^3$. $f$ takes vector $\textbf{x}=[x_1 \; x_2 \; x_3]$ and maps it to a scalar. Now, is $f$ a function or a functional? Thanks!

I was reading wikipedia about the short comings of the object oriented programing paradigm and a prof atCarnegie Mellon University states ""This semester Dan Licata and I are co-teaching a new course on functional programming for first-year prospective CS majors... Object-oriented programming...

I'm looking for a Real Analysis book to start with, besides Spivak. On Amazon, one of the reviewers said it was good as a subsequent book for learning Functional Analysis/Lebesque Integration, while another said it was a good introduction to Real Analysis. For those of you that have read it...

Hi guys, I am new to the forum. I have done a bit of reading on functional analysis lately.So I was wondering whether Functional Analysis can be related to physics in any way and what are the applications of that in physics?

Hi guys, I need some help please! Consider the following expression: \left[1-\int_{x}^{1}F(\rho(\xi))f(\xi)d\xi\right]^{n-1} where F:[0,1]\rightarrow [0,1] is a continuously differentiable function with F'=f, x∈[0,1], and n>2. Suppose that \rho belongs to the set of continuous and...

I am facing the following interesting question. A closed room\hall contains several identical machines in it, they are fed by an electrical cable. The machines can be turned on or off. When a machine is turned on, it consumes electrical energy and as a by product generates heat. The heat...

Hi all! I'm sorry if this question has been already asked in another post... I'm studying the path integrals formalism in QED. I'm dealing with the functional generator for fermionic fields. My question is: The generating functional is: $$Z_0=e^{-i\int{d^4xd^4y \bar{J}(x)S(x-y)J(y)}}$$...

If $f(x+y) = f(xy)$ and $\displaystyle f\left(-\frac{1}{2}\right) = -\frac{1}{2}$. Then $f(2012) = $

f(x) = e^{x} f(-x) with f(x) > 0 Is there anything I can say about the general shape of this function (defined on the real axis)? For example the formula gives the derivative of f in zero in terms of f(0) (which is okay assuming I'm only interested in f up to a multiplicative constant).

Hi all, I have recently been reading the book ``The Method of Second Quantization'' by Felix Berezin but I got trapped on just page 4, where the concept of generating functionals is introduced. It seems to be assigning each (anti-) symmetric function of N variables with a functional of a...

Homework Statement find the functional groups in the following compound: Homework Equations C8H16O4The Attempt at a Solution I know there is an ether but there is also something else. What is it? I have tried to find a group that is in the compound, but I have had no luck. It seems to be only...

If $f$ is a Real valued function on the set of real no. such that for any real $a$ and $b$ and $f(af(b)) = ab$. Then $f(2012) = $

I just finished my undergrad in Chem Eng and am very interested in energy field ( I was thinking of doing something with battery and storage systems as it is closest to my field). I was recently offered a masters in MSE for possibly working on battery materials using Density Functional Theory...

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Homework Statement 1. Given an operator H , and a sequence \{ H_n \} _{n\geq 1 } in an arbitrary Hilbert Space , such that both H and H_n are self-adjoint . How can I prove that if ||(H_n+i)^{-1} - (H+i) ^ {-1} || \to 0 and if H has an isolated eigenvalue \lambda of multiplicity...

Homework Statement A function from R-->R is differentiable and follows f( (x+y)/3 ) =( 2 + f(x) + f(y) ) / 3 Derivative of f(x) at x=2 is 2 Find the range of f ( |x| ) Homework Equations The Attempt at a Solution Well the questions asks me the range of f( |x| ). But i don't...

Hey all, I have been reading up on Green Functions and I stumbled upon the term "linear functional". I know the properties of the linear operator, but i can't really grasp what a functional does. In my notes it says that it indicates a linear function whose domain is a function space, and...

Hi, I have a question about a functional derivative. When determining the condition that the functional derivative have a stationary value of 0, do I use a partial derivative or a regular derivative? I would really appreciate the help. Thank you! David

Homework Statement i am doing project on gold nanoparticles i was reading a per regarding synthesis of gold nanoparticles. then i came across a statement "GNPs bearing functional moieties, which are anchored with thiol-linkers, in their monolayers" i didn't understand the word functional...

I've been watching Sidney Coleman's QFT lectures (http://www.physics.harvard.edu/about/Phys253.html). I've gotten up to his discussion of functional integration, and I have some questions. He starts out by discussing a finite-dimensional integral of a Gaussian function: \int{\frac{d^n...

I have a question regarding functional differentiablility. I understand that Frechet differntiability of a functional T with respect to a norm \rho_1 implies Hadamard differentiability of the functional T with respect to the same norm. However, it is no surprise that there would be cases...

Hello! On p.424 in the second edition of Altland & Simons, they compute an average as part of a renormalization group analysis of a model for dissipative quantum tunneling. I'd like to use their result in another situation, but I would be much happier if I understood how they derived it. I have...

Homework Statement I'm reading Kittel&kroemer's Thermal physics. How can I know Entropy's functional dependency? Author assume entropy's functional dependecy without explanations and derive some equaltities. So I can't follow it. N the number of particles. U Energy of the system. V. volume of...

As a basic exercise in C++ functional programming, I wrote the following code: #include <iostream> #include <string> using namespace std; template <class T> void Wib (T& a,T& b) { T temp = a; a = b; b = temp; }; int main() { string A = "World!"; string B = "Hello, "; Wib <string>...

To be specific, with total derivative I mean the linear map that best approximates a given function f at a given point. For f:ℝ\toℝ we have D(f,x_0):ℝ\toℝ, i.e. D(f,x_0)(h) \in ℝ. Often it is also denoted as just \delta f. Now in physics, in particular in the area of the Lagrangian, I find...

I know function is just a subset of functional but physical example helps to understand this difference.."Any physical situation" thanks in advance

Suppose $f(x)$ is continuous for all $x$ and $f(a+b)=f(a)+f(b)$ for all $a$ and $b$. Prove that $f(x)=Cx$, where $C=f(1)$. I have shown that $f(x)=Cx$ for all rational numbers. How do I use the continuity of $f$ to show it is true for all $x$?

I'm looking for a rigorous introduction to functional analysis in the style of Apostol. I've looked at Introductory Functional Analysis with Applications by Kreyszig, but I find it slightly too conversational. I know that Rudin has a Functional Analysis book, but it seems to be out of print...

Homework Statement S[y] = \int21dx ln(1 + xny'), y(1) = 1, y(2) = 21-n where n > 1 is a constant integer, and y is a continuously differentiable function for 1 ≤ x ≤ 2. Let h be a continuously differentiable function for 1 ≤ x ≤ 2 and ε a constant. Let ∆ = S[y + εh] − S[y]. Show that...

Homework Statement Let E be a dense linear subspace of a normed vector space X, and let Y be a Banach space. Suppose T0 \in £(E, Y) is a bounded linear operator from E to Y. Show that T0 can be extended to T\in £(E, Y) (by continuity) without increasing its norm. The Attempt at a Solution...

Is an open normed subspace Y (subset of X) primarily defined as a set {y in X : Norm(y) < r}? Where r is some real (positive) number. I know the open ball definitions and such... but it seems like this definition is saying, an open normed space, is essentially an open ball which satisfies...

the functional is \int_0^b{\frac{f(b)-f(x)}{b-x}}, . Another thing is it has defined endpoints. f(0)=0 and f(b) \in (0,10) and also f'(x)>0. oh and b \in (0,\infty) The euler-lagrange equation didn't give me anything helpful (not sure why..). One thing that is clear when looking at...

Alright, so I feel kind of dumb...but: I have been working on some QFT material, specifically derivation of Feynman rules for some more simple models ( \phi^{4} for example), and I have been seriously failing with functional derivatives. Every time I try to use the definition I mess up...

Hello! I was wondering how I could find the following derivatives from the given function using Jacobian determinants. f(u,v) = 0 u = lx + my + nz v = x^{2} + y^{2} + z^{2} \frac{∂z}{∂x} = ? (I believe y is constant, but the problem does not specify) \frac{∂z}{∂y} = ? (I...

Folks Are there any introductory functional analysis books which show calculus examples to illustrate the different axioms? thanks

Folks, Is there a difference between l subscript infinity and l superscript infinity. I believe the latter is the space if bounded sequences? thanks

Hi,all Is there someone's research related to DFT? I'm an undergraduate trying to get into it. I hope I could get some help hear if I have any question about that! Thanks a lot,Euphemia

"Applied Functional Analysis" by Zeidler In my book, "Applied Functional Analysis" by Zeidler, there's a question in the first chapter which, unless I got my concept of density wrong, I can't seem to see true : Let X=C[a,b] be the space of continuous functions on [a,b] with maximum norm. Then...

Homework Statement J(f)=\int 2xf−f′2+3f2f′dx f(0)=0,f(1)=−1. Homework Equations Ff-\frac{d}{dx}Ff'=0 The Attempt at a Solution Ff=2x+6f f'' Ff'=-2f' + 6f2 Plugging in, I get: 2x+6f f''- [itex]\frac{d}{dx} (-2f' + 6f2) 2x+6f f''-12f f'-2f''=0 Which doesn't look...

Homework Statement Let V be a finite dimensional vector field over F. Let T:V→V Let c be a scalar and suppose there is v in V such that T(v)=cv, then show there exists a non-zero linear functional f on V such that Tt(f)=cf. Tt denotes T transpose. Homework Equations Tt(f)=f°T...