Mfrhtyj

1 I'm currently on a theorem showing a general representation of linear functionals on the space of continuous functions on the interval $[0,1]$ . My problem is on the beginning of the proof. First we define a linear functional $f(x)$ on the space $C[0,1]$ . After that using the Cantor theorem we show that every continuous function on a compact interval is actually bounded and thus $C[0,1]$ is a subspace of $M[0,1]$ - bounded functions on the interval $[0,1]$ . Using the Hahn-Banach theorem we show that our functional $f$ can be extended to a functional $F$ on the whole space $M[0, 1]$ with the same norm. Maybe I've missed something foundamental, but how do we know that the functional $f$ is bounded, so that it has a norm and it's norm is the same with the extension $F$ ? Thanks in advance