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. D

    Understanding Cauchy Sequences in Banach Spaces

    Homework Statement http://img394.imageshack.us/img394/5994/67110701dt0.png Homework Equations A banach space is a complete normed space which means that every Cauchy sequence converges. The Attempt at a Solution I'm stuck at exercise (c). Suppose (f_n)_n is a Cauchy sequence in E. Then...
  2. J

    Proving C(X) is a Banach Space for Compact Hausdorff X

    I already know how to prove that if M is a compact metric space, then C(M) = \{f\in \mathbb{C}^M\;|\; f\;\textrm{continuous}\} with the sup-norm, is a Banach space, but now I encountered a claim, that actually metric is not necessary, and C(X) is Banach space also when X is a compact...
  3. D

    Is it possible for a Banach Space to be isomorphic to its double dual

    But the cannonical injection to not be an isomorphism?
  4. quasar987

    Bijection between Banach spaces.

    [SOLVED] Bijection between Banach spaces. Homework Statement Let E and F be two Banach space, f:E-->F be a continuous linear bijection and g:E-->F be linear and such that g\circ f^{-1} is continuous and ||g\circ f^{-1}||<1. Show that (f+g) is invertible and (f+g)^{-1} is continuous. [Hint...
  5. S

    How do i find k in the banach fixed point theorem

    how do i find k in the banach fixed point theorem. so say i have a function f(x)=1+3x-x^2 in the interval [1,2] then how do i find k? thank you
  6. S

    Banach fixed point theorem

    I really don't understand nothing from the Banach fixed point theorem, i know that it should satisfy: [g(x)-g(y)]<K(x-y) for all x and y in[a,b] but i don't even understand what that's supposed to mean? any help will be appreciated. thank you.
  7. H

    Banach Spaces vs. Closed Spaces: What's the Difference?

    Hi to all What exactly is the difference between Banach(=complete, as far as I understand) (sub)space and closed (sub)space. Is there a normed vector space that is complete but not closed or normed vectore space that is closed but not complete? Thanks in advance for explanation and/or examples.
  8. quasar987

    How can I show that the continuous dual X' of a normed space X is complete?

    [SOLVED] The continuous dual is Banach Homework Statement I'm trying to show that the continuous dual X' of a normed space X over K = R or C is complete. The Attempt at a Solution I have shown that if f_n is cauchy in X', then there is a functional f towards which f_n converge pointwise...
  9. J

    Projections on Banach and Hilbert spaces

    I've now encountered two different definitions for a projection. Let X be a Banach space. An operator P on it is a projection if P^2=P. Let H be a Hilbert space. An operator P on it is a projection if P^2=P and if P is self-adjoint. But the Hilbert space is also a Banach space, and there's...
  10. A

    X x Y is a Banach Space: Conditions Explored

    Hey there, could you guide me in the following question: X x Y is a Banach space if and only if X and Y are both Banach Spaces Thank you
  11. S

    Hahn Banach theorem example

    I'm trying to understand the Hahn Banach theorem, that every bounded linear functional f on some subspace M of a normed linear space X can be extended to a linear functional F on all of X with the same norm, and which agrees with f on M. But the proof is non-constructive, using zorn's lemma...
  12. C

    Differentiation on banach spaces

    Let L(A;B) be the space of linear maps l:A\rightarrow B. My goal is to derive the Leibniz (Product) Rule using the chain rule. Let f_i:U\subset E\rightarrow F_i, i=1,2 be differentiable maps and let B\in L(F_1,F_2;G). Then the mapping B(f_1,f_2)=B\circ (f_1\times f_2):U\subset E\rightarrow G...
  13. dextercioby

    Proving Noncompactness of Unit Ball in Infinite-Dimensional Banach Space

    Is the fact that a unit ball in an infinite-dimensional (Banach) space is a noncompact topological space...? If it is, how would one go about proving it...? Daniel.
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