# are linear functionals on C[0, 1] bounded and thus continuous

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 $$F$$?

No, there exist linear functionals on $$C[0,1]$$ that are not bounded.