1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Bijection between Banach spaces.

  1. Feb 14, 2008 #1


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    [SOLVED] Bijection between Banach spaces.

    1. The problem statement, all variables and given/known data
    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 [tex]g\circ f^{-1}[/tex] is continuous and [tex]||g\circ f^{-1}||<1[/tex]. Show that (f+g) is invertible and [tex](f+g)^{-1}[/tex] is continuous. [Hint: consider [tex](f+g)\circ f^{-1}[/tex]]

    3. The attempt at a solution

    The second part is easy once we have the first:

    1° g is continuous: let x be in E and y be the unique element of F such that y=f(x). Then, [tex]||g(x)||=||(g\circ f^{-1})(y)|| \leq ||g\circ f^{-1}||||y||<||f(x)||\leq||f||||x||[/tex] whence [tex]||g||\leq ||f||[/tex], i.e. g is continuous, since f is.

    2° It follows that f+g is continuous. Assuming it is also bijective, it is an open mapping, according to the open mapping theorem. That is to say, [tex](f+g)^{-1}[/tex] is continuous.

    But what about the first part?

    Based on the hint, it is obviously enough to show that [tex](f+g)\circ f^{-1}[/tex] is injective. (Since it is a mapping from F to F, it will then automatically be bijective , and composing this bijection with the bijection f yields (the bijection) f+g)

    But how do we show that [tex](f+g)\circ f^{-1} = \mathbb{I}_F+(g\circ f^{-1})[/tex] is injective?

    Note that I have used the fact that g o f^-1 is continuous, but I haven't really used the fact that its norm is strictly lesser than one. This amount to saying that g o f^-1 is a contraction. Since F is complete, by Banach's fixed point thm, it possesses a unique fixed point y*.
    Last edited: Feb 14, 2008
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted