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Math Help - Linear Functional

  1. #1
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    Linear Functional

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

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

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

    Any help would be great, Thanks
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  2. #2
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    Quote Originally Posted by ejgmath View Post
    I need to show that a linear functional \alpha_v on a Hilbert space H defined by \alpha_v(w)=<w,v> has operator norm \|\alpha_v\|_{op}=\|v\|.

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

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

    Any help would be great, Thanks
    Tip, plug in  v/||v||.
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  3. #3
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    Into where, can you elabourate?
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  4. #4
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    Quote Originally Posted by ejgmath View Post
    Into where, can you elabourate?
    Into  \alpha_v. What you want is \alpha_v(x)=||v|| for some x with ||x|| \leq 1, which precisely shows the inequality you are looking for (as ||\alpha_v||:=\sup \{|\alpha_v(x)|: ||x|| \leq 1 \} ).
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