Functional Definition and 380 Threads

Suppose that $$X$$ is a f-space and $$Y$$ is a subspace of $$X$$ and $$Y^{c}$$ is a first category in $$X$$. Can we show $$Y=X$$?

Hello I was doing an exercise that said: "If $P$ is a continuous operator in a Hilbert space $H$ and $P^2=P$ then the following five statements are equivalent". The first statement was that P is an orthogonal projection. Now this was suposed to be equivalent, under the condition of $P^2=P$, to...

Hi all, A professor asked me to do something, but I'm not quite sure what he means -- He asked me to use Density Functional Theory (DFT) calculations of the band structure of a certain crystalline metal and adjust the matrix elements of a Tight Binding (TB) model to make a "minimal" TB model...

Consider the functional Tf = f(5) - i f(7). If we take the domain T to be C_0(ℝ) with supremum norm, is T a bounded linear functional? What if we take the domain to be C_c(ℝ) with L^2 norm || . ||_2?I know I should post what I have so far but this time I have no idea because I had to missed 2...

Lets define trace for each square matrix A its trace as sum of its diagonal elements, so tr_{n}(A)=\sum_{j=1}^{n}a_{j,j}. Now proove that trace is a linear functional for all square matrix. I would be happy to know what has to be true for anything to be a linear functional? If I...

Homework Statement Find all the polynomials P(x) for which P(x^2+2x+3)=[P(x+3)]^2 Homework Equations The Attempt at a Solution I don't really know how to solve functional equations systematically. I tried to to find a linear P(x) and found P(x)=x-2 through trial & error. I also tried...

I know complex analysis is of immense help in physics at it aids us in calculating certain integrals much more easily. But what about real analysis and functional analysis? Are these branches of mathematical analysis of much use in physics? If so, in what branches of physics and how?

I'm reading about path integrals in Peskin and Schroeder's Introduction to Quantum field theory and there is a few things in the text which I find puzzling. At page 283 in the section about correlation functions we are considering the object (equation 9.15) \int D\phi(x) \phi(x_1) \phi(x_2)...

Homework Statement as some of you might've done it this is the functional eqUATION FROM THE IMO 2012 / a + b + c = 0 f2(a)+f2(b)+f2(c)=2f(a)f(b)+2f(b)f(c)+2f(c)f(a). f:Z->Z http://www.cut-the-knot.org/arithmetic/algebra/2012IMO-4.shtml <- link of the problem and its SOLUTION now i worked with...

SOLVED Prove that the functional sequence has no uniformly convergent subsequence -check $$n \in \mathbb{R}, \ \ f_n \ : \ \mathbb{R} \rightarrow \mathbb{R}, \ \ f_n(x) =\cos nx$$ We want to prove that $${f_n}$$ has no uniformly convergent subsequence. This is my attempt at proving that...

Hey all, back with another question. I have the opportunity in the fall to choose 1 (maybe 2 if I'm lucky) of the following classes: Numerical analysis (undergrad numerical linear algebra, using matlab), Functional Analysis (as a directed study course with a prof), and the other is doing a...

Homework Statement F[y(x)]=\int [y(x)\frac{dy(x)}{dx}+y(x)^{2}]\,dx Homework Equations δ(x-x') I think this is the Kronecker Delta. It might be the Dirac Delta. The Attempt at a Solution I have the whole thing written in my notes, I just don't know how to make sense of it...

Here's an example from my homework. I already turned it in, though. I basically just copied what I could from my notes, but I have no idea how this is done. Could someone explain this to me? I can't find anything intelligible (at least to me) of this stuff on any website. My notes contain parts...

Hi folks ... I urgently need good books about Functional analysis and Topology. These must be comprehensive and thorough, undergraduate or graduate. Please, advise and provide your experiences with such books. I accept only thick books ;) e.g Introductory Functional Analysis with...

Author: Michael Reed, Barry Simon Title: Functional Analysis Amazon link https://www.amazon.com/dp/0125850506/?tag=pfamazon01-20 Level: Undergrad

Author: Essential Results of Functional Analysis Title: Robert Zimmer Amazon Link: https://www.amazon.com/dp/0226983382/?tag=pfamazon01-20

Author: Frigyes Riesz, Bela Sz.-Nagy Title: Functional Analysis Amazon link: https://www.amazon.com/dp/0486662896/?tag=pfamazon01-20

I don't know what to title this but will change it if $f(f(x))$ has a name. Anyway, I need to find $f(x)$ such that $f(f(x))=-x$. My friend gave me this challenge question and I haven't been able to figure it out. There are many examples where $f(f(x))=x$ for example f(x)=\frac{1}{x} but that...

Author: Elias Stein, Rami Shakarchi Title: Functional Analysis: Introduction to Further Topics in Analysis Amazon Link: https://www.amazon.com/dp/0691113874/?tag=pfamazon01-20 Prerequisities: Real Analysis by Stein and Shakarchi Level: Undergrad Table of Contents: Foreword Introduction...

Author: Serge Lang Title: Real and Functional Analysis by Lang Amazon Link: https://www.amazon.com/dp/0387940014/?tag=pfamazon01-20 Prerequisities: Undergrad analysis Level: Grad Table of Contents: General Topology Sets Some Basic Terminology Denumerable Sets Zorn's Lemma...

I am a new user for Quantum Espresso(QE). Recently I have installed Quantum Espresso in my system. Now i am struggling to give input file in QE. How to generate input file in QE?

Author: Erwin Kreyszig Title: Introductory Functional Analysis wih Applications Amazon link https://www.amazon.com/dp/0471504599/?tag=pfamazon01-20 Prerequisities: Being acquainted with proofs and rigorous mathematics. Rigorous Calculus and Linear algebra. Level: Undergrad Table of...

I've always been curious why points in polar coordinates are defined as (r,θ) when all equations (including parametric equations formed from them) are defined as r=f(θ). Considering that point in cartesian coordinates are defined as (x,y) where y=f(x). Also a,b=(r,θ) ∫1/2[f(θ)]2 further...

A few days ago on MMF the following question was posted with no one showing how to solve it so far: Given: $\displaystyle f(f(x))=x^2-x+1\, \forall x\in \mathbb{R}$ find $\displaystyle f(x)$. I have never known how to solve such equations, except by trial and error, and this one has me...

Hi, I am trying to write down the propagator for a scalar field theory, but I want to try and get it in the functional representation. My plan is to compute the following: \langle \psi (x', t') | \psi (x,t) \rangle which gives the amplitude to go from x' to x. Now I guess I have to...

I have a definite integral defined by \begin{equation}T\left(G\left(g\right)\right)=\int_{g_{1}}^{g_{2}}G(g)\mathrm{d}g\end{equation} where G is a continuous function of a variable g, and g_{1} and g_{2} are known numbers. I want to minimize T\left(G\left(g\right)\right), that is I want to...

Homework Statement Hey, can I just check these functional derivatives?: 1) \frac{\delta F[g]}{\delta g(y)} where F[g] = \int dx \left[ \frac{1}{\sqrt{1+(g'(x))^2}} - 2g(x) + 5 \right]\>. 2) \frac{\delta F[a,b,g]}{\delta g(y)} where F[a,b,g] = \int d^4x \left[ A(\partial_{\mu}...

Homework Statement Determine whether the following functional series is pointwise and/or uniformly convergent: \sum_{n=1}^\infty \frac{x}{n} (x\in\mathbb{R}) Homework Equations The Attempt at a Solution My answer to this seems very straightforward and I would be very grateful if...

Homework Statement Let C be a non-empty convex subset of a real normed space (X,\|\cdot\|). Denote H(f,a):=\{x\in X: f(x)\leq a\} for f\in X^* (dual space) and a\in\mathbb{R}. Show that the closure \bar{C} of C satisfies \bar{C}=\bigcap_{f\in X^*,a\in\mathbb{R}: C\subseteq H(f,a)}H(f,a)...

Current follows the path of least resistance or shortest path. I just want to prove this or rather reproduce it using calculus of variations. I just want to show it in a fancy way. I want help to form the FUNCTIONAL for it. Useful equations: I=dq/dt=nqvA R=rho*l/A Where v is drift velocity...

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...