What is Banach: Definition and 63 Discussions

Banach is a Polish-language surname of several possible origins. Notable people with this surname include:

Stefan Banach (1892–1945), Polish mathematician
Ed Banach (born 1960), American wrestler
Lou Banach (born 1960), American wrestler
Korneliusz Banach (born 25 January 1994), Polish volleyball player
Łukasz Banach, birth name of Norman Leto (born 1980), Polish artist in the fields of painting, film, and new media
Maurice Banach, German footballer
Michael Banach (born 1962), American archbishop of the Roman Catholic Church
Orest Banach, German-American soccer goalkeeper of Ukrainian descent
William Banach (1903–1951), American politician, member of the Wisconsin State Assembly

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  1. C

    Help Needed Proving Implication for Linear Functional on Banach Space

    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...
  2. S

    MHB Understanding Example from Topics in Banach Space Integration

    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
  3. G

    I Vsauce's video on the Banach-Tarski paradox

    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...
  4. M

    MHB Banach fixed-point theorem : Existence of solution

    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...
  5. M

    Show that a space is a Banach space

    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...
  6. Rabindranath

    A Lagrange multipliers on Banach spaces (in Dirac notation)

    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...
  7. M

    MHB Proving that $X$ is a Banach Space and $Y$ is Not

    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...
  8. SemM

    A What separates Hilbert space from other spaces?

    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...
  9. J

    A Convergence of a cosine sequence in Banach space

    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...
  10. JulienB

    Maximum norm and Banach fixed-point theorem

    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...
  11. G

    I Physical implications from Vitali sets or Banach-Tarski?

    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...
  12. C

    Homeomorphism in a Banach space

    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...
  13. C

    An automorphism in a Banach space

    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...
  14. Domenico94

    A Holder space is a Banach space

    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...

    Proof of Banach Lemma: Small Matrix Eigenvalues

    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?
  16. O

    MHB Generalization of Banach contraction principle

    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...
  17. O

    MHB Contraction constant in banach contraction principle

    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...
  18. O

    MHB Why is the contraction constant important in the Banach contraction principle?

    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...
  19. T

    Motivations for the C*-algebra of observables?

    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...
  20. DavideGenoa

    Banach space as Banach 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...
  21. Fredrik

    Banach Sub-Algebra of C*-Algebra: Proving Completeness

    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...
  22. K

    MHB Matrix norm in Banach space

    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 \\...
  23. C

    Hilbert, Banach and Fourier theory

    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...
  24. P

    Why Banach Spaces Are Important: A Fellow Student's Question

    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...
  25. M

    Sum of two closed subspaces in a Banach space

    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...
  26. S

    Quantum Gravity: Lie Groups vs. Banach Algebras & Spectral Theory

    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 &...
  27. A

    Spectrum of a linear operator on a Banach space

    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...
  28. MathematicalPhysicist

    G's Guide to GR on Banach Spaces

    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
  29. F

    Does the compact subset of an infinite Banach have finite span?

    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...
  30. Z

    Spectrum of a linear operator on a Banach space

    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...
  31. T

    Compactness of sets in Banach spaces

    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...
  32. T

    Compactness/convergence in Banach spaces

    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?
  33. S

    Creating convergent sequences in Banach spaces

    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...
  34. T

    How are Banach and Hilbert spaces applied in quantum mechanics?

    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?
  35. C

    MHB Proving T is Continuous in a Complex Banach Space

    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?
  36. U

    Embedding L1 in the Banach space of complex Borel measures

    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...
  37. C

    Banach Fixed Point and Differential Equations

    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...
  38. R

    Prove that a normed space is not Banach

    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?
  39. MathematicalPhysicist

    Books on differential geometry on Banach Spaces.

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

    Using Correlation :Random Variables as a Normed Space (Banach, Hilbert maybe.?)

    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...
  41. Fredrik

    If A is a Banach algebra, then so is A/I

    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...
  42. N

    Cosine = Contraction? (Banach)

    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?)
  43. T

    Spectrum, banach algebra

    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
  44. L

    Problem on seperable banach spaces

    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...
  45. N

    Complete Norm on M_n(R): Questions & Hints

    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...
  46. M

    Is p(X) closed in X** for a Banach space X?

    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...
  47. D

    Explore Banach Spaces and Bounded Linear Operators

    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...
  48. B

    Is the Closure of a Subset in l^{1} Compact?

    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.
  49. C

    Banach Space Problem: Proving Subspaces Contain e-Orthogonal Elements

    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...
  50. O

    Banach Space that is NOT Hilbert

    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!