Banach Definition and 62 Threads

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...

But the cannonical injection to not be an isomorphism?

[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...

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

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.

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.

[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...

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...

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

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...

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...

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.