Functional analysis Definition and 111 Threads

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

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...

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...

Homework Statement Let (X,\|\cdot\|) be a reflexive Banach space. Let \{T_n\}_{n\in\mathbb{N}} be a sequence of bounded linear operators from X into X such that \lim_{n\to\infty}f(T_nx) exists for all f\in X' and x\in X. Use the Uniform Boundedness Principle (twice) to show that...

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)...

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...

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?

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...

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.

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?

Homework Statement Let \lambda_1 ,..., \lambda_n be the eigenvalues of an nXn self-adjoint matrix A, written in increasing order. Show that for any m \leq n one has: \sum_{r=1}^{m} \lambda_r = min \{ tr(L) :dim(L) =m \} where L denotes any linear subspace of \mathbb {C} ^n , and...

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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 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...

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

"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...

Folks, I am starting a module in functional analysis undergrad level. I have been suggested introductory functional analysis by Kreyszig, but in instead of buying another expensive book is there a good online source like a pdf on in this topic that I could avail of? Any help will be...

A functional analysis' problem I hope this is the right place to submit this post. Homework Statement Let A be a symmetric operator, A\supseteq B and \mathcal{R}_{A+\imath I}=\mathcal{R}_{B+\imath I} (where \mathcal{R} means the range of the operator). Show that A=B. 2. The attempt at a...

Hello everybody here, I'm taking Functional Analysis this term, and the textbook is : "An Introduction to Hilbert Space, Cambridge, 1988" by N. Young. Unfortunately, we have to solve most of the book's problems. So, does anyone has some of them ? I found a list of solved problems on...

My professor tried to show the following in lecture the other day: If T is a linear operator on a Hilbert space and (Tz,z) is real for every z in H, then T is bounded and self-adjoint. Below, I use (*,*) to indicate the Hilbert space inner product. He told us to use the identity (which I've...

Hello, I'm reading through John Conway's A Course in Functional Analysis and I'm having trouble understanding example 1.5 on page 168 (2nd edition): Let (X, \Omega, \mu) and M_\phi : L^p(\mu) \to L^p(\mu) be as in Example III.2.2 (i.e., sigma-finite measure space and M_\phi f = \phi f is a...

Homework Statement I want to understand the proof of proposition 7.1 in Conway. The theorem says that if \{P_i|i\in I\} is a family of projection operators, and P_i is orthogonal to P_j when i\neq j, then for any x in a Hilbert space H, \sum_{i\in I}P_ix=Px where P is the projection...

In Random Operators in Fixed point theory of functional analysis, Is there any relation between the saparable space and measurable functions?,, what are the random operators?

Can anyone tell me the Engineering applications of Functional analysis with a real world example, If possible? Thanks in advance.

Are there any theorems concerning this expression \frac{1}{z}f\left(\frac{1}{z}\right). I appreciate posts of any theorems you can think of.

If X is Banach space and F:X \rightarrow X is a linear and bounded map and that F^n(x)\rightarrow0 pointwise .. How can I show that it converges to zero uniformly also? Thanks

We know that a linear operator T:X\rightarrowY between two Banach Spaces X and Y is an open mapping if T is surjective. Here open mapping means that T sends open subsets of X to open subsets of Y. Prove that if T is an open mapping between two Banach Spaces then it is not necessarily a closed...

Does anybody know of any good resources for this? Specifically for real analysis, I'm looking for something that covers calculus on manifolds, differential forms, Lebesgue integration, etc. and for functional analysis: metric spaces, Banach spaces, Hilbert spaces, Fourier series, etc. Thanks!

I'm going to be applying to grad schools next year (I have an undergrad degree in math and phyisics), and I have narrowed down my areas of interest to two fields: functional analysis and it's involvement in QFT; and computational/theoretical neuroscience. I find pure math more enjoyable, but I'm...

Hi PF, I am currently trying to teach myself the rudiments of differential forms, in particular their application to physics, and there's something I'd like to ask. It seems like diff forms can be used to express all kinds of physics, but the area I haven't been able to figure out is stuff...

Where can I get a very basic introduction to the current research directions in functional analysis? I have done a basic course in it. Also I am interested in knowing about applications of Ramsey theory to functional analysis. Thanks.

Could any of you recommend a functional analysis textbook? I have looked at "Methods of modern mathematical physics" by Reed&Simon, but they assume a pure-maths BSc background, thus this book is not ideal for me. About my background: I have an Applied Physics BSc and starting a Theoretical...

Homework Statement Show that the range \mathcal{R}(T) of a bounded linear operator T: X \rightarrow Y is not necessarily closed. Hint: Use the linear bounded operator T: l^{\infty} \rightarrow l^{\infty} defined by (\eta_{j}) = T x, \eta_{j} = \xi_{j}/j, x = (\xi_{j}). Homework Equations...

This is from Rudin, Functional Analysis 2.1. Not homework. If X is an infinite-dimensional topological vector space which is the union of countably many finite-dimensional subspaces, prove X is first category in itself. What about this example? Take R^n (standard n-dimensional space of...

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I had a quick question on a part of a proof in chapter 1 of Functional Analysis, by Professor Rudin. Theorem 1.10 states "Suppose K and C are subsets of a topological vector space X. K is compact, and C is closed, and the intersection of K and C is the empty set. Then 0 has a...

I'm thinking about getting this book. I'm a physics major, and I think the only analysis course I'm required to take later as a prerequisite for graduate courses is Introduction to Complex Analysis. So far, I've taken Cal I-III and Linear Algebra. Differential Equations will probably be in the...

Hello, I am currently looking for a book on functional analysis. However most books I have seen assume knowledge real and complex analysis. But I am looking for a more superficial introduction covering the important results, some examples of applications (mainly to computational problems)...

I'm looking for a entry-level book discussing the application of functional analysis to differential equations- mostly the Navier-Stokes equation, but PDEs in general. The books I have or have seen are either math books, full of proofs and definitions without application, or physics papers...

Homework Statement http://img357.imageshack.us/img357/8695/38808719uw6.png Homework Equations \lim_n a_n := \lim_{n \rightarrow \infty} a_n The Attempt at a Solution I'm stuck at exercise (c). Since if n heads to infinity the m doesn't play the role the limit must be one. So...

Does anyone know of where I should look to find lots of good functional analysis problems? I am currently reading Kreyszig which has great commentary, but the majority of the exercises are simple.

[SOLVED] Sequences in lp spaces... (Functional Analysis) Homework Statement Find a sequence which converges to zero but is not in any lp space where 1<=p<infinity. Homework Equations N/A The Attempt at a Solution I strongly suspect 1/ln(n+1) is a solution. Since ln(n+1) ->...

Hello I need help with an analysis proof and I was hoping someone might help me with it. The question is: Let (X,d) be a metric space and say A is a subset of X. If x is an accumulation point of A, prove that every r-neighbourhood of x actually contains an infinite number of distinct...