Banach Definition and 63 Threads

Dear everybody, I am having some trouble proving the implication (or the forward direction.) Here is my work: Suppose that we have an arbitrary linear functional ##l## on a Banach Space ##B## is continuous. Since ##l## is continuous linear functional on B, in other words, we want show that...

Hey Could you give me a hint how to explain this example? Need help to prove statement in red frame. Example from book (Topics In Banach Space Integration) by Ye Guoju‏، Schwabik StefanThank you

A question to Vsauce's famous video about the Banach-Tarski paradox at 10:09: Can you really construct the hyper-webster like that? If you choose the order like that, you'll never get any words containing other letters than "A". Shouldn't you choose an order like A, ... , Z, AA, ..., AZ...

Hey! :o We have the system \begin{align*}&x_1=\left (5+x_1^2+x_2^2\right )^{-1} \\ &x_2=\left (x_1+x_2\right )^{\frac{1}{4}}\end{align*} and the set $G=\{\vec{x}\in \mathbb{R}^2: \|\vec{x}-\vec{c}\|_{\infty}\leq 0.2\}$ where $\vec{c}=(0.2,1)^T$. I want to show with the Banach fixed-point...

Homework Statement Show the following space equipped with given norm is a Banach space. Let ##C^k[a,b]## with ##a<b## finite and ##k \in \mathbb{N}## denote the set of all continuous functions ##u:[a,b]\to \mathbb R## that have continuous derivatives on ##[a,b]## to order ##k##. Define the...

I want to prove Cauchy–Schwarz' inequality, in Dirac notation, ##\left<\psi\middle|\psi\right> \left<\phi\middle|\phi\right> \geq \left|\left<\psi\middle|\phi\right>\right|^2##, using the Lagrange multiplier method, minimizing ##\left|\left<\psi\middle|\phi\right>\right|^2## subject to the...

Hey! :o Let $ T> 0 $ be fixed. We denote $ X = \{f \in C (\mathbb{R}): f (t) = f (t + T) \ \forall t \in \mathbb {R} \} $ and $ Y = \{f \in C^1 (\mathbb{R}): f (t) = f (t + T) \ \forall t \in \mathbb {R} \} $ be the spaces of the $ T $ periodic continuous and continuously differentiable...

Hi, I have the impression that the special thing about Hilbert space for Quantum Mechanics is that it is simply an infinite space, which allows for infinitively integration and derivation of its elements, f(x), g(x), their linear combination, or any other complex function, given that the main...

Does the sequence \{f_n\}=\{\cos{(2nt)}\} converge or diverge in Banach space C(-1,1) endowed with the sup-norm ||f||_{\infty} = \text{sup}_{t\in (-1,1)}|f(t)| ? At first glance my intuition is that this sequence should diverge because cosine is a period function. But how to really prove...

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...

Hi. Can we infer something about physics from stuff like Vitali sets or the Banach-Tarski paradox? Maybe if we assume the energy in a given space volume to be well defined and finite, that there must be fundamental particles that can't be split, or that there must be a Planck length and energy...

Hello, 1. Homework Statement Let be E a banach space, A a continuous automorphsim(by the banach theorem his invert is continus too.). and f a k lipshitzian fonction with $$k < \frac{1}{||A^{-1}||}$$. Homework Equations $$k < \frac{1}{||A^{-1}||}$$ The Attempt at a Solution I have to show...

Hello I've got a problem : let be a normed vectorial space E, N and A an continue automorphism. I suppose E is complete. So by the banach theorem $$A^{−1}$$ is continue. So now let be f a k lipshitz application with $$k<\frac{1}{||A^{−1}||}$$. . I'd like to show that f + A is an...

Hi everyone. I was just reading Evans' book on PDE, and, at some point, it asked to prove that an holder space is a Banach space, and I tried to do that. I just want to ask you if my proof is correct (if you see dumb errors, just notice also that I study EE, so I'm not much into doing proofs...

Hi, I found the following relationship in a proof for gradient of log det x $$(I+A)^{-1}=I-A$$ When A is a "small" matrix (?? eigenvalues) I am not sure how to prove it, any ideas?

Let $\left(X,d\right)$ be a complete metric space and suppose that $f:X\to X$ satisfies $d\left(fx,fy\right)\le\beta\left(d\left(x,y\right)\right)d\left(x,y\right)$ for all x,y $\in$ X where $\beta$ is a decreasing function on ${R}^{+}$ to $[0,1)$. Then $f$ has a unique fixed point. The...

İn some fixed point theory books, I saw an expression...But I didnt understand what this mean...Please can you help me ? " It was important in the proof of banach contraction principle that the contraction constant "h" be strictly less than 1. Than gave us control over the rate of convergence...

It was important in the proof of BCP that the contraction constant h be strictly less than 1. That gave us control over the rate of convergence of f^n (x0) to the fixed point since h^n goes to 0 as n goes to infinity...If we consider f is contractive mapping instead of a contraction, then we...

As far as I understand, the Hilbert space formalism can be derived using functional analysis and representation theory (not familiar with those) from the requirement that observables (their mathematical models) form a C*-algebra and the possible states of a system map the members of the algebra...

I find, in Kolmogorov-Fomin's Элементы теории функций и функционального анализа, at the end of § 5 of chapter IV, several statement on the spectral radius and the non-emptyness of the spectrum of a linear operator ina Banach space, which are left without proof. Nevertheless, in Tikhomirov's...

Edit: I originally wrote that ##\mathcal A## is a Banach algebra. The assumption that goes into the theorem is stronger. It's a C*-algebra. I am however still mainly interested in the claim that ##\mathcal A_1##, as defined below, is a Banach sub-algebra of ##\mathcal B(\mathcal A)##. Let...

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. I want to get a quick overview of the theory of Hilbert spaces in order to understand Fourier series and transforms at a higher level. I have a couple of questions I hoped someone could help me with. - First of all: Can anyone recommend any literature, notes etc.. which go through the...

A fellow student of mine asked a question to our teacher in functional analysis, and the answer we got was not very satisfactory. In our discussion on Banach spaces the student asked "Why is it interesting/important for a normed space to be complete?". To my surprise the teacher said something...

Homework Statement . Let ##E## be a Banach space and let ##S,T \subset E## two closed subspaces. Prove that if dim## T< \infty##, then ##S+T## is also closed. The attempt at a solution. To prove that ##S+T## is closed I have to show that if ##x## is a limit point of ##S+T##, then ##x \in...

Quantum Gravity: "Lie Groups" vs. "Banach Algebras & Spectral Theory" I'm interested in researching quantum gravity & non-commutative geometry. I am planning to take one math course outside of my physics classes this Fall to help, but can't decide between two: "Lie groups" or "Banach algebras &...

I'm trying to understand the spectrum and resolvent of a linear operator on a Banach space in as much generality as I possibly can. It seems that the furthest the concept can be "pulled back" is to a linear operator T: D(T) \to X, where X is a Banach space and D(T)\subseteq X. But here are a...

OK, I started reading GR for mathematicians from Wu and Sachs. And I see that from the start that they look on finite dimensional linear algebra, has there been any treatment for a general setting? MP

Homework Statement Hi all, I am struggling with getting an intuitive understanding of linear normed spaces, particularly of the infinite variety. In turn, I then am having trouble with compactness. To try and get specific I have two questions. Question 1 In a linear normed vector space, is...

Homework Statement I have a number of problems, to be completed in the next day or so (!) that I am pretty stuck with where to begin. They involve calculating the spectra of various different linear operators. Homework Equations The first was: Let X be the space of complex-valued...

Homework Statement Working in a banach space (X,\|\cdot\|) we have a sequence of compact sets A_k\subset X. Assume that there exist r_k>0 such that \sum_{k\in\mathbb{N}}r_k<\infty and for every k\in\mathbb{N}: $$A_{k+1}\subset\{x+u|x\in A_k,u\in X,\|u\|\leq r_k\}.$$Prove that the closure of...

Been doing exercises on compactness/sequential compactness of objects in Banach spaces and some of my solutions come down to whether holds in a) arbitrary finite-dimensional Banach space b) lp, 1 <= p <= infinity Does it?

Sorry for the rather vague title! Homework Statement Given: Two Banach spaces A and B, and a linear map T: A\rightarrow B The sequences (x^n_i) in A. For each fixed n, (x^n_i) \rightarrow 0 for i \rightarrow \infty. The sequences (Tx^n_i) in B. For each fixed n, (Tx^n_i) \rightarrow y_n...

As a european bachelor student in physics, i can follow a theoretical math course next year about banach and hilbert spaces. How useful are those subjects for physics?

Let X be a complex Banach space and T in L(X,X) a linear operator. Assuming only that (T*f)(x)=f(Tx), where x in X and f in X* how can I prove that T is continuous?

Hey, I know this is commonly a homework question, but it came up in my own studies; so this isn't a homework question for me. I hope it's alright that I put it here. I'm trying to show that if f dx = d\lambda for some f \in L^1(\mathbb{R}^d) and complex Borel measure \lambda then |f| dx...

Homework Statement Find the value of x, correct to three decimal places for which: \int^{x}_{0}\frac{t^{2}}{1+t^{2}}dt=\frac{1}{2}. Homework Equations Banach's Fixed Point Theorem Picard's Theorem? The Attempt at a Solution I'm not sure where to start with this type of problem...

Hello everyone, I have a problem and cannot solve it. Could you help? Here it is We have a normed space and an uncountable Hamel basis of it. Prove that it is not a Banach space. Should I use Baire theorem? Any suggestions?

Can you recommend me of books or preprints that cover reasonabely well this topic? Thanks.

Hi, everyone: I have been curious for a while about the similarity between the correlation function and an inner-product: Both take a pair of objects and spit out a number between -1 and 1, so it seems we could define a notion of orthogonality in a space of random variables, so...

Homework Statement The problem is to prove the following: If \mathcal A is a Banach algebra, and \mathcal I is a closed ideal in \mathcal A, then \mathcal A/\mathcal I is a Banach algebra. This is problem 3.1.3 (4)(b) in "Functional analysis: spectral theory", by V.S. Sunder. Link. Homework...

So in Analysis I we explained the convergence of cos to a fixed value by Banach's contraction theorem. But is the cos a strict contraction? Is that obvious? (What is its contraction factor?)

Homework Statement How to show element of finite dimensional banach algebra has finite spectrum? Homework Equations spectrum(x) = set of complex numbers 'c' with cI-x not invertible, I is identity The Attempt at a Solution please help to start, I don't know

Prove that if a Banach space X, has separable dual X*, then X is separable. It gives the hint that the first line of the proof should be to take a countable dense subset \{f_n\} of X* and choose x_n\in X such that for each n, we have ||x_n||=1 and |f_n(x)|\geq(1/2)||f_n||. Ok so what do I...

Let M_n(R) be the n x n matrices over the reals R. Define a norm || || on M_n(R) by ||A||= sum of absolute values of all the entries of A. Further define a new norm || ||* by ||A||* = sup{||AX||/||X||, ||X||!=0}. Show that 1. M_n(R) under || ||* is complete. 2. If ||A||<1, then I-A is...

Let X be a Banach space. We show p(X) is closed in X**, where p:X-->X** is defined by p(x)=T_x and T_x:X*-->F is defined by T_x(x*)=x*(x) (F is a field). I think I should pick a convergent sequence {x_n} in p(X) (x_n -->x)and show that x belongs to p(X). i.e. show there exists a w in X such...

Homework Statement http://img252.imageshack.us/img252/4844/56494936eo0.png 2. relevant equations BL = bounded linear space (or all operators which are bounded). The Attempt at a Solution I got for the first part: ||A||_{BL} =||tf(t)||_{\infty} \leq ||f||_{\infty} so ||A||_{BL} \leq 1...

Homework Statement Consider the Banach Space l^{1}. Let S={x \in l^{1}|\left\|x\right\|<1}. Is S a compact subset of l^{1}? prove or Disprove.

Homework Statement Let E be a Banach space and let M be a closed subspace of E. A vector x in E is called e-orthogonal to M if for all y in M the following inequality holds: ||x+y||>= (1- e)||x||. Prove that for each e>0 any proper subspace of M contains e-orthogonal elements...

I know that all Hilbert spaces are Banach spaces, and that the converse is not true, but I've been unable to come up with a (hopefully simple!) example of a Banach space that is not also a Hilbert space. Any help would be appreciated!