Results 1 to 4 of 4

Thread: Linear Functional

  1. #1
    Junior Member
    Joined
    Jan 2010
    Posts
    32

    Linear Functional

    I need to show that a linear functional $\displaystyle \alpha_v$ on a Hilbert space $\displaystyle H$ defined by $\displaystyle \alpha_v(w)=<w,v>$ has operator norm $\displaystyle \|\alpha_v\|_{op}=\|v\|$.

    I have already shown by the Cauchy Schwarz inequality that $\displaystyle \|\alpha_v\|_{op}\leq\|v\|$.

    I think now I need to show that $\displaystyle \|\alpha_v\|_{op}\geq\|v\|$.

    Any help would be great, Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member Focus's Avatar
    Joined
    Aug 2009
    Posts
    228
    Quote Originally Posted by ejgmath View Post
    I need to show that a linear functional $\displaystyle \alpha_v$ on a Hilbert space $\displaystyle H$ defined by $\displaystyle \alpha_v(w)=<w,v>$ has operator norm $\displaystyle \|\alpha_v\|_{op}=\|v\|$.

    I have already shown by the Cauchy Schwarz inequality that $\displaystyle \|\alpha_v\|_{op}\leq\|v\|$.

    I think now I need to show that $\displaystyle \|\alpha_v\|_{op}\geq\|v\|$.

    Any help would be great, Thanks
    Tip, plug in $\displaystyle v/||v||$.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Jan 2010
    Posts
    32
    Into where, can you elabourate?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member Focus's Avatar
    Joined
    Aug 2009
    Posts
    228
    Quote Originally Posted by ejgmath View Post
    Into where, can you elabourate?
    Into $\displaystyle \alpha_v$. What you want is $\displaystyle \alpha_v(x)=||v||$ for some x with $\displaystyle ||x|| \leq 1$, which precisely shows the inequality you are looking for (as $\displaystyle ||\alpha_v||:=\sup \{|\alpha_v(x)|: ||x|| \leq 1 \}$ ).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linear functional
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: Sep 1st 2011, 03:55 AM
  2. linear functional inner product
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Nov 15th 2010, 10:20 PM
  3. Linear Functional
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: Feb 4th 2010, 05:40 PM
  4. Linear functional
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: May 11th 2009, 10:29 AM
  5. not continuous linear functional
    Posted in the Calculus Forum
    Replies: 0
    Last Post: Jan 13th 2009, 12:33 PM

Search Tags


/mathhelpforum @mathhelpforum