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T:H-->F is a bounded linear functional..where F=R(real) or C(complex) Now,||T(x)||=||T||_op (operator norm) now what can be told about ||x||??

My guess it should be ||x||<=1.but also i can't prove this nor i can't find such example that there exist x such that ||x||>1 and ||T(x)||=||T||.. so any help will be appreciated..

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    $\begingroup$ Hello start here and learn how to properly format your question. Many people will "pass" on your question for ones that are easier to read. It is not hard. Good luck! $\endgroup$
    – rschwieb
    May 10 at 18:57

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We have $|T(x)|\leq ||T|| ||x||$. So if we have $|T(x_0)|=||T||$ for some $x_0$ we must have $1 \leq ||x_0||$.

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