# Linear Functionals - Continuity and Boundedness

## Homework Statement

Prove that a continuous linear functional, $$f$$ is bounded and vice versa.

## Homework Equations

I know that the definition of a linear functional is:
$$f( \alpha|x> + \beta|y>) = \alpha f(|x> ) + \beta f( |y> )$$
and that a bounded linear functional satisfies:
$$||f(|x>)) || \leq \epsilon ||\ |x> || , \ \ \ \epsilon > 0$$

## The Attempt at a Solution

I tried the following by letting:

$$f(|x>) = \sum a_{i}f( |x_{i}> )$$
then applying triangle inequality:
$$|| f(|x>) || \leq \sum||a_{i}|| \ || f( |x_{i}> ) ||$$