Results 1 to 4 of 4

Math Help - showing continuity

  1. #1
    Newbie
    Joined
    Jan 2011
    Posts
    24

    showing continuity

    Hello!

    I have a problem. Somehow it doesn't seem all that tricky but I just can't get my head around it. Ok, here it goes: Let X, Y, Z be Banach spaces and let m:X\times Y\rightarrow Z be a map satisfying: for fixed x in X m(-,y) is linear and continuous (and the same for a fixed y in Y). How do I show that my map m is continuous aswell.

    First I was thinking that I should just simply show that it is bdd using the continuity in the two cases but I can't really manage.

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32

    Re: showing continuity

    Do you know the principle of uniform boundedness?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jan 2011
    Posts
    24

    Re: showing continuity

    Yes I do. Is that the trick?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32

    Re: showing continuity

    You have to show that m is bounded on the unit ball. Put S_X=\left\{x\in X, ||x||=1\right\}. Then \{m(x,\cdot)\}_{x\in S_X} is a family of elements of Y^*. Since the map x\mapsto m(x,y) is linear and continuous for each y, the set \{m(x,y)\}_{x\in S_X} is bounded for each fixed y. So \{||m(x,\cdot)||\} is bounded and you can conclude.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. showing continuity
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: September 6th 2011, 03:53 PM
  2. showing continuity of of limit function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: November 17th 2010, 07:56 PM
  3. Showing(2)
    Posted in the Trigonometry Forum
    Replies: 3
    Last Post: January 23rd 2010, 08:48 PM
  4. Continuity, Uniform Continuity
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 1st 2009, 09:36 PM
  5. showing f'(-x)=f'(x)
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 26th 2007, 09:14 AM

Search Tags


/mathhelpforum @mathhelpforum