Results 1 to 6 of 6

Math Help - Continuity and norm space

  1. #1
    Junior Member
    Joined
    May 2009
    Posts
    54

    Continuity and norm space

    Let V a norma space with 2 norms , lets say norm 1 and norm 2

    Show that the function identity $\ I_d :\left( {V,\left\| x \right\|_1 } \right) \to \left( {V,\left\| x \right\|_2 } \right)\$ es continuos iff the set
    A \A = \left\{ {x \in V/\left\| x \right\|_1  = 1} \right\}\$ is bounded with norm 2

    Please somebody give me a biiiiiiiiiiiiig hint....

    Regards...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by orbit View Post
    Let V a norma space with 2 norms , lets say norm 1 and norm 2

    Show that the function identity $\ I_d :\left( {V,\left\| x \right\|_1 } \right) \to \left( {V,\left\| x \right\|_2 } \right)\$ es continuos iff the set
    A \A = \left\{ {x \in V/\left\| x \right\|_1  = 1} \right\}\$ is bounded with norm 2

    Please somebody give me a biiiiiiiiiiiiig hint....

    Regards...
    Come on man--you really need to show some effort. You need remember that linear operators are continuous if and only if they're bounded. So, \text{id}:V\to V will be bounded if and only if there is a constant M such that \|\text{id}(v)\|_1\leqslant M\|v\|_2 for all v\in V... now why is this equivalent to saying that this inequality holds true for every v\in B_{\|\cdot\|_1}(0;1)?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    May 2009
    Posts
    54
    Quote Originally Posted by Drexel28 View Post
    Come on man--you really need to show some effort. You need remember that linear operators are continuous if and only if they're bounded. So, \text{id}:V\to V will be bounded if and only if there is a constant M such that \|\text{id}(v)\|_1\leqslant M\|v\|_2 for all v\in V... now why is this equivalent to saying that this inequality holds true for every v\in B_{\|\cdot\|_1}(0;1)?
    Hi, thanl you for answering. unfortunately for me,linear operators havenīt been taught is my course.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by orbit View Post
    Hi, thanl you for answering. unfortunately for me,linear operators havenīt been taught is my course.
    Ok, so then what have you been taught?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    May 2009
    Posts
    54
    Quote Originally Posted by Drexel28 View Post
    Ok, so then what have you been taught?
    Here is what I`ve been taught.
    Attached Files Attached Files
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by orbit View Post
    Here is what I`ve been taught.
    I don't mean to be rude but

    A) I'm not going to read a whole book to help you with one problem.


    B) That's in Spanish man.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Norm on dual space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: June 2nd 2011, 02:16 AM
  2. Norm on the Lp space
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 20th 2010, 07:15 PM
  3. norm and normed space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 24th 2009, 02:32 AM
  4. Prove that the norm on L^2 space is continuous
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 27th 2009, 01:16 AM
  5. vector space and norm space
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: January 3rd 2009, 04:51 PM

Search Tags


/mathhelpforum @mathhelpforum