# T:H-->F is a bounded linear functional..where F=R(real) or C(complex) Now,||T(x)||=||T||_op (operator norm) this is it be true that ||x||>1??

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