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## Homework Statement

Prove that a continuous linear functional, [tex] f [/tex] is bounded and vice versa.

## Homework Equations

I know that the definition of a linear functional is:

[tex] f( \alpha|x> + \beta|y>) = \alpha f(|x> ) + \beta f( |y> ) [/tex]

and that a bounded linear functional satisfies:

[tex] ||f(|x>)) || \leq \epsilon ||\ |x> || , \ \ \ \epsilon > 0[/tex]

## The Attempt at a Solution

I tried the following by letting:

[tex] f(|x>) = \sum a_{i}f( |x_{i}> ) [/tex]

then applying triangle inequality:

[tex] || f(|x>) || \leq \sum||a_{i}|| \ || f( |x_{i}> ) || [/tex]

but now I'm stuck, can someone please help get going in right direction? thanks!