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!
Best Answer
No, there exist linear functionals on $C[0,1]$ that are not bounded.
In fact, for every infinite dimensional space there exists a discontinuous (and therefore unbounded) linear functional, see here.
I would suspect that there is an additional assumption somewhere that the linear functional is bounded.